Separated setsFrom Wikipedia, the free encyclopedia

Chapter 1

Connected space

For other uses, see Connection (disambiguation).Connected and disconnected subspaces of R²

A

B

C

D

E4

E1

E2

E3

From top to bottom: red space A, pink space B, yellow space C and orange space D are all connected, whereas greenspace E (made of subsets E1, E2, E3, and E4) is not connected. Furthermore, A and B are also simply connected(genus 0), while C and D are not: C has genus 1 and D has genus 4.

In topology and related branches of mathematics, a connected space is a topological space that cannot be representedas the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topologicalproperties that is used to distinguish topological spaces. A stronger notion is that of a path-connected space, whichis a space where any two points can be joined by a path.

2

1.1. FORMAL DEFINITION 3

A subset of a topological space X is a connected set if it is a connected space when viewed as a subspace of X.An example of a space that is not connected is a plane with an infinite line deleted from it. Other examples ofdisconnected spaces (that is, spaces which are not connected) include the plane with an annulus removed, as well asthe union of two disjoint closed disks, where all examples of this paragraph bear the subspace topology induced bytwo-dimensional Euclidean space.

1.1 Formal definition

A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, Xis said to be connected. A subset of a topological space is said to be connected if it is connected under its subspacetopology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article doesnot follow that practice.For a topological space X the following conditions are equivalent:

1. X is connected.

2. X cannot be divided into two disjoint nonempty closed sets.

3. The only subsets of X which are both open and closed (clopen sets) are X and the empty set.

4. The only subsets of X with empty boundary are X and the empty set.

5. X cannot be written as the union of two nonempty separated sets (sets for which each is disjoint from the other’sclosure).

6. All continuous functions from X to {0,1} are constant, where {0,1} is the two-point space endowed with thediscrete topology.

1.1.1 Connected components

The maximal connected subsets (ordered by inclusion) of a nonempty topological space are called the connectedcomponents of the space. The components of any topological space X form a partition of X: they are disjoint,nonempty, and their union is the whole space. Every component is a closed subset of the original space. It followsthat, in the case where their number is finite, each component is also an open subset. However, if their number isinfinite, this might not be the case; for instance, the connected components of the set of the rational numbers are theone-point sets, which are not open.Let Γx be the connected component of x in a topological space X, and Γ′

x be the intersection of all clopen setscontaining x (called quasi-component of x.) Then Γx ⊂ Γ′

x where the equality holds if X is compact Hausdorff orlocally connected.

1.1.2 Disconnected spaces

A space in which all components are one-point sets is called totally disconnected. Related to this property, a space Xis called totally separated if, for any two distinct elements x and y of X, there exist disjoint open neighborhoods Uof x and V of y such that X is the union of U and V. Clearly any totally separated space is totally disconnected, butthe converse does not hold. For example take two copies of the rational numbers Q, and identify them at every pointexcept zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering thetwo copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the conditionof being totally separated is strictly stronger than the condition of being Hausdorff.

1.2 Examples• The closed interval [0, 2] in the standard subspace topology is connected; although it can, for example, bewritten as the union of [0, 1) and [1, 2], the second set is not open in the chosen topology of [0, 2].

4 CHAPTER 1. CONNECTED SPACE

• The union of [0, 1) and (1, 2] is disconnected; both of these intervals are open in the standard topological space[0, 1) ∪ (1, 2].

• (0, 1) ∪ {3} is disconnected.

• A convex set is connected; it is actually simply connected.

• AEuclidean plane excluding the origin, (0, 0), is connected, but is not simply connected. The three-dimensionalEuclidean space without the origin is connected, and even simply connected. In contrast, the one-dimensionalEuclidean space without the origin is not connected.

• A Euclidean plane with a straight line removed is not connected since it consists of two half-planes.

• ℝ, The space of real numbers with the usual topology, is connected.

• If even a single point is removed from ℝ, the remainder is disconnected. However, if even a countable infinityof points are removed from ℝn, where n≥2, the remainder is connected.

• Any topological vector space over a connected field is connected.

• Every discrete topological space with at least two elements is disconnected, in fact such a space is totallydisconnected. The simplest example is the discrete two-point space.[1]

• On the other hand, a finite set might be connected. For example, the spectrum of a discrete valuation ringconsists of two points and is connected. It is an example of a Sierpiński space.

• The Cantor set is totally disconnected; since the set contains uncountably many points, it has uncountably manycomponents.

• If a space X is homotopy equivalent to a connected space, then X is itself connected.

• The topologist’s sine curve is an example of a set that is connected but is neither path connected nor locallyconnected.

• The general linear group GL(n,R) (that is, the group of n-by-n real, invertible matrices) consists of two con-nected components: the one with matrices of positive determinant and the other of negative determinant.In particular, it is not connected. In contrast, GL(n,C) is connected. More generally, the set of invertiblebounded operators on a (complex) Hilbert space is connected.

• The spectra of commutative local ring and integral domains are connected. More generally, the following areequivalent[2]

1. The spectrum of a commutative ring R is connected2. Every finitely generated projective module over R has constant rank.3. R has no idempotent ̸= 0, 1 (i.e., R is not a product of two rings in a nontrivial way).

1.3 Path connectedness

A path from a point x to a point y in a topological space X is a continuous function f from the unit interval [0,1] toX with f(0) = x and f(1) = y. A path-component of X is an equivalence class of X under the equivalence relationwhich makes x equivalent to y if there is a path from x to y. The space X is said to be path-connected (or pathwiseconnected or 0-connected) if there is exactly one path-component, i.e. if there is a path joining any two points inX. Again, many authors exclude the empty space.Every path-connected space is connected. The converse is not always true: examples of connected spaces that arenot path-connected include the extended long line L* and the topologist’s sine curve.However, subsets of the real lineR are connected if and only if they are path-connected; these subsets are the intervalsofR. Also, open subsets ofRn orCn are connected if and only if they are path-connected. Additionally, connectednessand path-connectedness are the same for finite topological spaces.

1.4. ARC CONNECTEDNESS 5

This subspace of R² is path-connected, because a path can be drawn between any two points in the space.

1.4 Arc connectedness

A space X is said to be arc-connected or arcwise connected if any two distinct points can be joined by an arc, thatis a path f which is a homeomorphism between the unit interval [0, 1] and its image f([0, 1]). It can be shown anyHausdorff space which is path-connected is also arc-connected. An example of a space which is path-connected butnot arc-connected is provided by adding a second copy 0' of 0 to the nonnegative real numbers [0, ∞). One endowsthis set with a partial order by specifying that 0'<a for any positive number a, but leaving 0 and 0' incomparable.One then endows this set with the order topology, that is one takes the open intervals (a, b) = {x | a < x < b} and thehalf-open intervals [0, a) = {x | 0 ≤ x < a}, [0', a) = {x | 0' ≤ x < a} as a base for the topology. The resulting spaceis a T1 space but not a Hausdorff space. Clearly 0 and 0' can be connected by a path but not by an arc in this space.

1.5 Local connectedness

Main article: Locally connected space

A topological space is said to be locally connected at a point x if every neighbourhood of x contains a connectedopen neighbourhood. It is locally connected if it has a base of connected sets. It can be shown that a space X islocally connected if and only if every component of every open set of X is open. The topologist’s sine curve is anexample of a connected space that is not locally connected.Similarly, a topological space is said to be locally path-connected if it has a base of path-connected sets. An opensubset of a locally path-connected space is connected if and only if it is path-connected. This generalizes the earlierstatement about Rn and Cn, each of which is locally path-connected. More generally, any topological manifold islocally path-connected.

1.6 Set operations

The intersection of connected sets is not necessarily connected.

6 CHAPTER 1. CONNECTED SPACE

AB

A B

A

B

A

Bconnexenon connexe

intersection intersection

connexe non connexe

union union

Examples of unions and intersections of connected sets

The union of connected sets is not necessarily connected. Consider a collection {Xi} of connected sets whose unionisX = ∪iXi . IfX is disconnected and U ∪ V is a separation ofX (with U, V disjoint and open inX ), then eachXi must be entirely contained in either U or V , since otherwise, Xi ∩ U and Xi ∩ V (which are disjoint and openin Xi ) would be a separation of Xi , contradicting the assumption that it is connected.This means that, if the union X is disconnected, then the collection {Xi} can be partitioned to two sub-collections,such that the unions of the sub-collections are disjoint and open inX (see picture). This implies that in several cases,a union of connected sets is necessarily connected. In particular:

1. If the common intersection of all sets is not empty ( ∩Xi ̸= ∅ ), then obviously they cannot be partitioned tocollections with disjoint unions. Hence the union of connected sets with non-empty intersection is connected.

2. If the intersection of each pair of sets is not empty ( ∀i, j : Xi∩Xj ̸= ∅ ) then again they cannot be partitionedto collections with disjoint unions, so their union must be connected.

3. If the sets can be ordered as a “linked chain”, i.e. indexed by integer indices and ∀i : Xi ∩Xi+1 ̸= ∅ , thenagain their union must be connected.

4. If the sets are pairwise-disjoint and the quotient space X/{Xi} is connected, then X must be connected.Otherwise, if U ∪ V is a separation of X then q(U) ∪ q(V ) is a separation of the quotient space (sinceq(U), q(V ) are disjoint and open in the quotient space).[3]

1.6. SET OPERATIONS 7

Each ellipse is a connected set, but the union is not connected, since it can be partitioned to two disjoint open sets U and V.

Two connected sets whose difference is not connected

The set difference of connected sets is not necessarily connected. However, if X⊇Y and their difference X\Y isdisconnected (and thus can be written as a union of two open sets X1 and X2), then the union of Y with each suchcomponent is connected (i.e. Y∪Xi is connected for all i). Proof:[4] By contradiction, suppose Y∪X1 is not connected.So it can be written as the union of two disjoint open sets, e.g. Y∪X1 = Z1∪Z2. Because Y is connected, it must be

8 CHAPTER 1. CONNECTED SPACE

entirely contained in one of these components, say Z1, and thus Z2 is contained in X1. Now we know that:

X = (Y∪X1)∪X2 = (Z1∪Z2)∪X2 = (Z1∪X2)∪(Z2∩X1)

The two sets in the last union are disjoint and open in X, so there is a separation of X, contradicting the fact that X isconnected.

1.7 Theorems

“Main theorem of connectedness” redirects to here.

• Main theorem: Let X and Y be topological spaces and let f : X → Y be a continuous function. If X is (path-)connected then the image f(X) is (path-)connected. This result can be considered a generalization of theintermediate value theorem.

• Every path-connected space is connected.

• Every locally path-connected space is locally connected.

• A locally path-connected space is path-connected if and only if it is connected.

• The closure of a connected subset is connected.

• The connected components are always closed (but in general not open)

• The connected components of a locally connected space are also open.

• The connected components of a space are disjoint unions of the path-connected components (which in generalare neither open nor closed).

• Every quotient of a connected (resp. locally connected, path-connected, locally path-connected) space is con-nected (resp. locally connected, path-connected, locally path-connected).

• Every product of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).

• Every open subset of a locally connected (resp. locally path-connected) space is locally connected (resp. locallypath-connected).

• Every manifold is locally path-connected.

1.8 Graphs

Graphs have path connected subsets, namely those subsets for which every pair of points has a path of edges joiningthem. But it is not always possible to find a topology on the set of points which induces the same connected sets. The5-cycle graph (and any n-cycle with n>3 odd) is one such example.As a consequence, a notion of connectedness can be formulated independently of the topology on a space. To wit,there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivityaxioms; their morphisms are those functions which map connected sets to connected sets (Muscat & Buhagiar 2006).Topological spaces and graphs are special cases of connective spaces; indeed, the finite connective spaces are preciselythe finite graphs.However, every graph can be canonically made into a topological space, by treating vertices as points and edges ascopies of the unit interval (see topological graph theory#Graphs as topological spaces). Then one can show that thegraph is connected (in the graph theoretical sense) if and only if it is connected as a topological space.

1.9. STRONGER FORMS OF CONNECTEDNESS 9

1.9 Stronger forms of connectedness

There are stronger forms of connectedness for topological spaces, for instance:

• If there exist no two disjoint non-empty open sets in a topological space, X, X must be connected, and thushyperconnected spaces are also connected.

• Since a simply connected space is, by definition, also required to be path connected, any simply connected spaceis also connected. Note however, that if the “path connectedness” requirement is dropped from the definitionof simple connectivity, a simply connected space does not need to be connected.

• Yet stronger versions of connectivity include the notion of a contractible space. Every contractible space ispath connected and thus also connected.

In general, note that any path connected space must be connected but there exist connected spaces that are not pathconnected. The deleted comb space furnishes such an example, as does the above-mentioned topologist’s sine curve.

1.10 See also• connected component (graph theory)

• Connectedness locus

• Extremally disconnected space

• locally connected space

• n-connected

• uniformly connected space

1.11 References

1.11.1 Notes[1] George F. Simmons (1968). Introduction to Topology and Modern Analysis. McGraw Hill Book Company. p. 144. ISBN

0-89874-551-9.

[2] Charles Weibel, The K-book: An introduction to algebraic K-theory

[3] Credit: Saaqib Mahmuud and Henno Brandsma at Math StackExchange.

[4] Credit: Marek at Math StackExchange

1.11.2 General references

• Munkres, James R. (2000). Topology, Second Edition. Prentice Hall. ISBN 0-13-181629-2.

• Weisstein, Eric W., “Connected Set”, MathWorld.

• V. I. Malykhin (2001), “Connected space”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Muscat, J; Buhagiar, D (2006). “Connective Spaces” (PDF). Mem. Fac. Sci. Eng. Shimane Univ., Series B:Math. Sc. 39: 1–13..

Chapter 2

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topologicalspace in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on atopological space, the “Hausdorff condition” (T2) is the most frequently used and discussed. It implies the uniquenessof limits of sequences, nets, and filters.Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff’s original definitionof a topological space (in 1914) included the Hausdorff condition as an axiom.

2.1 Definitions

U

x

V

y

The points x and y, separated by their respective neighbourhoods U and V.

Points x and y in a topological space X can be separated by neighbourhoods if there exists a neighbourhood U of xand a neighbourhood V of y such that U and V are disjoint (U ∩ V = ∅). X is a Hausdorff space if any two distinctpoints of X can be separated by neighborhoods. This condition is the third separation axiom (after T0 and T1), whichis why Hausdorff spaces are also called T2 spaces. The name separated space is also used.A related, but weaker, notion is that of a preregular space. X is a preregular space if any two topologically distin-guishable points can be separated by neighbourhoods. Preregular spaces are also called R1 spaces.The relationship between these two conditions is as follows. A topological space is Hausdorff if and only if it is bothpreregular (i.e. topologically distinguishable points are separated by neighbourhoods) and Kolmogorov (i.e. distinctpoints are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient is

10

2.2. EQUIVALENCES 11

Hausdorff.

2.2 Equivalences

For a topological space X, the following are equivalent:

• X is a Hausdorff space.

• Limits of nets in X are unique.[1]

• Limits of filters on X are unique.[2]

• Any singleton set {x} ⊂ X is equal to the intersection of all closed neighbourhoods of x.[3] (A closed neigh-bourhood of x is a closed set that contains an open set containing x.)

• The diagonal Δ = {(x,x) | x ∈ X} is closed as a subset of the product space X × X.

2.3 Examples and counterexamples

Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers (under the standardmetric topology on real numbers) are a Hausdorff space. More generally, all metric spaces are Hausdorff. In fact,many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff conditionexplicitly stated in their definitions.A simple example of a topology that is T1 but is not Hausdorff is the cofinite topology defined on an infinite set.Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in theconstruction of Hausdorff gauge spaces. Indeed, when analysts run across a non-Hausdorff space, it is still probablyat least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff.In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry,in particular as the Zariski topology on an algebraic variety or the spectrum of a ring. They also arise in the modeltheory of intuitionistic logic: every complete Heyting algebra is the algebra of open sets of some topological space,but this space need not be preregular, much less Hausdorff, and in fact usually is neither. The related concept of Scottdomain also consists of non-preregular spaces.While the existence of unique limits for convergent nets and filters implies that a space is Hausdorff, there are non-Hausdorff T1 spaces in which every convergent sequence has a unique limit.[4]

2.4 Properties

Subspaces and products of Hausdorff spaces are Hausdorff,[5] but quotient spaces of Hausdorff spaces need not beHausdorff. In fact, every topological space can be realized as the quotient of some Hausdorff space.[6]

Hausdorff spaces are T1, meaning that all singletons are closed. Similarly, preregular spaces are R0.Another nice property of Hausdorff spaces is that compact sets are always closed.[7] This may fail in non-Hausdorffspaces such as Sierpiński space.The definition of a Hausdorff space says that points can be separated by neighborhoods. It turns out that this impliessomething which is seemingly stronger: in a Hausdorff space every pair of disjoint compact sets can also be separatedby neighborhoods,[8] in other words there is a neighborhood of one set and a neighborhood of the other, such that thetwo neighborhoods are disjoint. This is an example of the general rule that compact sets often behave like points.Compactness conditions together with preregularity often imply stronger separation axioms. For example, any locallycompact preregular space is completely regular. Compact preregular spaces are normal, meaning that they satisfyUrysohn’s lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite opencovers. The Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, andevery compact Hausdorff space is normal Hausdorff.

12 CHAPTER 2. HAUSDORFF SPACE

The following results are some technical properties regarding maps (continuous and otherwise) to and fromHausdorffspaces.Let f : X → Y be a continuous function and suppose Y is Hausdorff. Then the graph of f, {(x, f(x)) | x ∈ X} , isa closed subset of X × Y.Let f : X → Y be a function and let ker(f) ≜ {(x, x′) | f(x) = f(x′)} be its kernel regarded as a subspace of X ×X.

• If f is continuous and Y is Hausdorff then ker(f) is closed.

• If f is an open surjection and ker(f) is closed then Y is Hausdorff.

• If f is a continuous, open surjection (i.e. an open quotient map) then Y is Hausdorff if and only if ker(f) isclosed.

If f,g : X→ Y are continuous maps and Y is Hausdorff then the equalizer eq(f, g) = {x | f(x) = g(x)} is closed inX. It follows that if Y is Hausdorff and f and g agree on a dense subset of X then f = g. In other words, continuousfunctions into Hausdorff spaces are determined by their values on dense subsets.Let f : X→ Y be a closed surjection such that f−1(y) is compact for all y ∈ Y. Then if X is Hausdorff so is Y.Let f : X→ Y be a quotient map with X a compact Hausdorff space. Then the following are equivalent

• Y is Hausdorff

• f is a closed map

• ker(f) is closed

2.5 Preregularity versus regularity

All regular spaces are preregular, as are all Hausdorff spaces. There are many results for topological spaces that holdfor both regular and Hausdorff spaces. Most of the time, these results hold for all preregular spaces; they were listedfor regular and Hausdorff spaces separately because the idea of preregular spaces came later. On the other hand,those results that are truly about regularity generally do not also apply to nonregular Hausdorff spaces.There are many situations where another condition of topological spaces (such as paracompactness or local compact-ness) will imply regularity if preregularity is satisfied. Such conditions often come in two versions: a regular versionand a Hausdorff version. Although Hausdorff spaces are not, in general, regular, a Hausdorff space that is also (say)locally compact will be regular, because any Hausdorff space is preregular. Thus from a certain point of view, itis really preregularity, rather than regularity, that matters in these situations. However, definitions are usually stillphrased in terms of regularity, since this condition is better known than preregularity.See History of the separation axioms for more on this issue.

2.6 Variants

The terms “Hausdorff”, “separated”, and “preregular” can also be applied to such variants on topological spacesas uniform spaces, Cauchy spaces, and convergence spaces. The characteristic that unites the concept in all of theseexamples is that limits of nets and filters (when they exist) are unique (for separated spaces) or unique up to topologicalindistinguishability (for preregular spaces).As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condi-tion in these cases reduces to the T0 condition. These are also the spaces in which completeness makes sense, andHausdorffness is a natural companion to completeness in these cases. Specifically, a space is complete if and only ifevery Cauchy net has at least one limit, while a space is Hausdorff if and only if every Cauchy net has at most onelimit (since only Cauchy nets can have limits in the first place).

2.7. ALGEBRA OF FUNCTIONS 13

2.7 Algebra of functions

The algebra of continuous (real or complex) functions on a compact Hausdorff space is a commutative C*-algebra,and conversely by the Banach–Stone theorem one can recover the topology of the space from the algebraic propertiesof its algebra of continuous functions. This leads to noncommutative geometry, where one considers noncommutativeC*-algebras as representing algebras of functions on a noncommutative space.

2.8 Academic humour• Hausdorff condition is illustrated by the pun that in Hausdorff spaces any two points can be “housed off” fromeach other by open sets.[9]

• In the Mathematics Institute of the University of Bonn, in which Felix Hausdorff researched and lectured,there is a certain room designated the Hausdorff-Raum. This is a pun, as Raum means both room and spacein German.

2.9 See also• Quasitopological space

• Weak Hausdorff space

• Fixed-point space, a Hausdorff space X such that every continuous function f:X→X has a fixed point.

2.10 Notes[1] Willard, pp. 86–87.

[2] Willard, pp. 86–87.

[3] Bourbaki, p. 75.

[4] van Douwen, Eric K. (1993). “An anti-Hausdorff Fréchet space in which convergent sequences have unique limits”.Topology and its Applications 51 (2): 147–158. doi:10.1016/0166-8641(93)90147-6.

[5] Hausdorff property is hereditary at PlanetMath.org.

[6] Shimrat, M. (1956). “Decomposition spaces and separation properties”. Quart. J. Math. 2: 128–129.

[7] Proof of A compact set in a Hausdorff space is closed at PlanetMath.org.

[8] Willard, p. 124.

[9] Colin Adams and Robert Franzosa. Introduction to Topology: Pure and Applied. p. 42

2.11 References• Arkhangelskii, A.V., L.S. Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4.

• Bourbaki; Elements of Mathematics: General Topology, Addison-Wesley (1966).

• Hazewinkel, Michiel, ed. (2001), “Hausdorff space”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

Chapter 3

Neighbourhood (mathematics)

For the concept in graph theory, see Neighbourhood (graph theory).In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts

A set V in the plane is a neighbourhood of a point p if a small disk around p is contained in V .

in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where one canmove that point some amount without leaving the set.

14

3.1. DEFINITION 15

pV

A rectangle is not a neighbourhood of any of its corners.

This concept is closely related to the concepts of open set and interior.

3.1 Definition

If X is a topological space and p is a point in X , a neighbourhood of p is a subset V of X that includes an openset U containing p ,

p ∈ U ⊆ V.

This is also equivalent to p ∈ X being in the interior of V .Note that the neighbourhood V need not be an open set itself. If V is open it is called an open neighbourhood.Some scholars require that neighbourhoods be open, so it is important to note conventions.A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets containingeach of its points.The collection of all neighbourhoods of a point is called the neighbourhood system at the point.If S is a subset ofX then a neighbourhood of S is a set V that includes an open set U containing S . It follows thata set V is a neighbourhood of S if and only if it is a neighbourhood of all the points in S . Furthermore, it followsthat V is a neighbourhood of S iff S is a subset of the interior of V .

3.2 In a metric space

In a metric space M = (X, d) , a set V is a neighbourhood of a point p if there exists an open ball with centre pand radius r > 0 , such that

16 CHAPTER 3. NEIGHBOURHOOD (MATHEMATICS)

A set S in the plane and a uniform neighbourhood V of S .

aa-ε a+ε

The epsilon neighbourhood of a number a on the real number line.

Br(p) = B(p; r) = {x ∈ X | d(x, p) < r}

is contained in V .V is called uniform neighbourhood of a set S if there exists a positive number r such that for all elements p of S ,

Br(p) = {x ∈ X | d(x, p) < r}

is contained in V .For r > 0 the r -neighbourhood Sr of a set S is the set of all points in X that are at distance less than r from S(or equivalently, S r is the union of all the open balls of radius r that are centred at a point in S ).It directly follows that an r -neighbourhood is a uniform neighbourhood, and that a set is a uniform neighbourhoodif and only if it contains an r -neighbourhood for some value of r .

3.3. EXAMPLES 17

M

( )a-ε a+εa

( [ )) [ ]

The set M is a neighbourhood of the number a, because there is an ε-neighbourhood of a which is a subset of M.

3.3 Examples

Given the set of real numbers R with the usual Euclidean metric and a subset V defined as

V :=∪n∈N

B (n ; 1/n) ,

then V is a neighbourhood for the set N of natural numbers, but is not a uniform neighbourhood of this set.

3.4 Topology from neighbourhoods

The above definition is useful if the notion of open set is already defined. There is an alternative way to define atopology, by first defining the neighbourhood system, and then open sets as those sets containing a neighbourhood ofeach of their points.A neighbourhood system onX is the assignment of a filter N(x) (on the set X ) to each x in X , such that

1. the point x is an element of each U in N(x)

2. each U in N(x) contains some V in N(x) such that for each y in V , U is in N(y) .

One can show that both definitions are compatible, i.e. the topology obtained from the neighbourhood system definedusing open sets is the original one, and vice versa when starting out from a neighbourhood system.

3.5 Uniform neighbourhoods

In a uniform space S = (X, δ) , V is called a uniform neighbourhood of P if P is not close toX \V , that is thereexists no entourage containing P and X \ V .

3.6 Deleted neighbourhood

A deleted neighbourhood of a point p (sometimes called a punctured neighbourhood) is a neighbourhood of p ,without {p} . For instance, the interval (−1, 1) = {y : −1 < y < 1} is a neighbourhood of p = 0 in the real line,so the set (−1, 0)∪ (0, 1) = (−1, 1) \ {0} is a deleted neighbourhood of 0 . Note that a deleted neighbourhood of agiven point is not in fact a neighbourhood of the point. The concept of deleted neighbourhood occurs in the definitionof the limit of a function.

3.7 See also

• Tubular neighborhood

18 CHAPTER 3. NEIGHBOURHOOD (MATHEMATICS)

3.8 References• Kelley, John L. (1975). General topology. New York: Springer-Verlag. ISBN 0-387-90125-6.

• Bredon, Glen E. (1993). Topology and geometry. New York: Springer-Verlag. ISBN 0-387-97926-3.

• Kaplansky, Irving (2001). Set Theory and Metric Spaces. American Mathematical Society. ISBN 0-8218-2694-8.

Chapter 4

Normal space

For normal vector space, see normal (geometry).

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4:every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called aT4 space. These conditions are examples of separation axioms and their further strengthenings define completelynormal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces.

4.1 Definitions

A topological space X is a normal space if, given any disjoint closed sets E and F, there are open neighbourhoodsU of E and V of F that are also disjoint. More intuitively, this condition says that E and F can be separated byneighbourhoods.

U

E

V

F

The closed sets E and F, here represented by closed disks on opposite sides of the picture, are separated by their respective neigh-bourhoods U and V, here represented by larger, but still disjoint, open disks.

A T4 space is a T1 space X that is normal; this is equivalent to X being normal and Hausdorff.A completely normal space or a hereditarily normal space is a topological space X such that every subspace of Xwith subspace topology is a normal space. It turns out that X is completely normal if and only if every two separatedsets can be separated by neighbourhoods.A completely T4 space, or T5 space is a completely normal T1 space topological space X, which implies that X is

19

20 CHAPTER 4. NORMAL SPACE

Hausdorff; equivalently, every subspace of X must be a T4 space.A perfectly normal space is a topological space X in which every two disjoint closed sets E and F can be preciselyseparated by a continuous function f fromX to the real lineR: the preimages of {0} and {1} under f are, respectively,E and F. (In this definition, the real line can be replaced with the unit interval [0,1].)It turns out that X is perfectly normal if and only if X is normal and every closed set is a Gδ set. Equivalently, X isperfectly normal if and only if every closed set is a zero set. Every perfectly normal space is automatically completelynormal.[1]

A Hausdorff perfectly normal space X is a T6 space, or perfectly T4 space.Note that the terms “normal space” and “T4" and derived concepts occasionally have a different meaning. (Nonethe-less, “T5" always means the same as “completely T4", whatever that may be.) The definitions given here are the onesusually used today. For more on this issue, see History of the separation axioms.Terms like “normal regular space" and “normal Hausdorff space” also turn up in the literature – they simply meanthat the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space isthe same thing as a T4 space. Given the historical confusion of the meaning of the terms, verbal descriptions whenapplicable are helpful, that is, “normal Hausdorff” instead of “T4", or “completely normal Hausdorff” instead of “T5".Fully normal spaces and fully T4 spaces are discussed elsewhere; they are related to paracompactness.A locally normal space is a topological space where every point has an open neighbourhood that is normal. Everynormal space is locally normal, but the converse is not true. A classical example of a completely regular locallynormal space that is not normal is the Nemytskii plane.

4.2 Examples of normal spaces

Most spaces encountered in mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces:

• All metric spaces (and hence all metrizable spaces) are perfectly normal Hausdorff;

• All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal regular, although not ingeneral Hausdorff;

• All compact Hausdorff spaces are normal;

• In particular, the Stone–Čech compactification of a Tychonoff space is normal Hausdorff;

• Generalizing the above examples, all paracompact Hausdorff spaces are normal, and all paracompact regularspaces are normal;

• All paracompact topological manifolds are perfectly normal Hausdorff. However, there exist non-paracompactmanifolds which are not even normal.

• All order topologies on totally ordered sets are hereditarily normal and Hausdorff.

• Every regular second-countable space is completely normal, and every regular Lindelöf space is normal.

Also, all fully normal spaces are normal (even if not regular). Sierpinski space is an example of a normal space thatis not regular.

4.3 Examples of non-normal spaces

An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on thespectrum of a ring, which is used in algebraic geometry.A non-normal space of some relevance to analysis is the topological vector space of all functions from the real lineR to itself, with the topology of pointwise convergence. More generally, a theorem of A. H. Stone states that theproduct of uncountably many non-compact metric spaces is never normal.

4.4. PROPERTIES 21

4.4 Properties

Every closed subset of a normal space is normal. The continuous and closed image of a normal space is normal.[2]

The main significance of normal spaces lies in the fact that they admit “enough” continuous real-valued functions, asexpressed by the following theorems valid for any normal space X.Urysohn’s lemma: If A and B are two disjoint closed subsets of X, then there exists a continuous function f from Xto the real line R such that f(x) = 0 for all x in A and f(x) = 1 for all x in B. In fact, we can take the values of f to beentirely within the unit interval [0,1]. (In fancier terms, disjoint closed sets are not only separated by neighbourhoods,but also separated by a function.)More generally, the Tietze extension theorem: If A is a closed subset of X and f is a continuous function from A toR, then there exists a continuous function F: X→ R which extends f in the sense that F(x) = f(x) for all x in A.If U is a locally finite open cover of a normal space X, then there is a partition of unity precisely subordinate to U.(This shows the relationship of normal spaces to paracompactness.)In fact, any space that satisfies any one of these conditions must be normal.A product of normal spaces is not necessarily normal. This fact was first proved by Robert Sorgenfrey. An example ofthis phenomenon is the Sorgenfrey plane. Also, a subset of a normal space need not be normal (i.e. not every normalHausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čechcompactification (which is normal Hausdorff). A more explicit example is the Tychonoff plank.

4.5 Relationships to other separation axioms

If a normal space is R0, then it is in fact completely regular. Thus, anything from “normal R0" to “normal completelyregular” is the same as what we normally call normal regular. Taking Kolmogorov quotients, we see that all normalT1 spaces are Tychonoff. These are what we normally call normal Hausdorff spaces.A topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, thereare disjoint open sets containing them. Every normal space is pseudonormal, but not vice versa.Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpinski spaceis normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.

4.6 Citations[1] Munkres 2000, p. 213

[2] Willard, Stephen (1970). General topology. Reading, Mass.: Addison-Wesley Pub. Co. pp. 100–101. ISBN 0486434796.

4.7 References• Kemoto, Nobuyuki (2004). “Higher Separation Axioms”. In K.P. Hart, J. Nagata, and J.E. Vaughan. Ency-clopedia of General Topology. Amsterdam: Elsevier Science. ISBN 0-444-50355-2.

• Munkres, James R. (2000). Topology (2nd ed.). Prentice-Hall. ISBN 0-13-181629-2.

• Sorgenfrey, R.H. (1947). “On the topological product of paracompact spaces”. Bull. Amer. Math. Soc. 53:631–632. doi:10.1090/S0002-9904-1947-08858-3.

• Stone, A. H. (1948). “Paracompactness and product spaces”. Bull. Amer. Math. Soc. 54: 977–982.doi:10.1090/S0002-9904-1948-09118-2.

• Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6.

Chapter 5

Separable space

Not to be confused with Separated space.

In mathematics a topological space is called separable if it contains a countable, dense subset; that is, there exists asequence {xn}∞n=1 of elements of the space such that every nonempty open subset of the space contains at least oneelement of the sequence.Like the other axioms of countability, separability is a “limitation on size”, not necessarily in terms of cardinality(though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtletopological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorffspace is determined by its values on the countable dense subset.Contrast separability with the related notion of second countability, which is in general stronger but equivalent on theclass of metrizable spaces.

5.1 First examples

Any topological space which is itself finite or countably infinite is separable, for the whole space is a countable densesubset of itself. An important example of an uncountable separable space is the real line, in which the rationalnumbers form a countable dense subset. Similarly the set of all vectors (r1, . . . , rn) ∈ Rn in which ri is rational forall i is a countable dense subset of Rn ; so for every n the n -dimensional Euclidean space is separable.A simple example of a space which is not separable is a discrete space of uncountable cardinality.Further examples are given below.

5.2 Separability versus second countability

Any second-countable space is separable: if {Un} is a countable base, choosing any xn∈Un from the non-empty Un

gives a countable dense subset. Conversely, a metrizable space is separable if and only if it is second countable, whichis the case if and only if it is Lindelöf.To further compare these two properties:

• An arbitrary subspace of a second countable space is second countable; subspaces of separable spaces need notbe separable (see below).

• Any continuous image of a separable space is separable (Willard 1970, Th. 16.4a).; even a quotient of a secondcountable space need not be second countable.

• A product of at most countably many separable spaces is separable. A countable product of second countablespaces is second countable, but an uncountable product of second countable spaces need not even be firstcountable.

22

5.3. CARDINALITY 23

5.3 Cardinality

The property of separability does not in and of itself give any limitations on the cardinality of a topological space: anyset endowed with the trivial topology is separable, as well as second countable, quasi-compact, and connected. The“trouble” with the trivial topology is its poor separation properties: its Kolmogorov quotient is the one-point space.A first countable, separable Hausdorff space (in particular, a separable metric space) has at most the continuumcardinality c . In such a space, closure is determined by limits of sequences and any convergent sequence has at mostone limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subsetto the points of X .A separable Hausdorff space has cardinality at most 2c , where c is the cardinality of the continuum. For this closureis characterized in terms of limits of filter bases: if Y ⊆ X and z ∈ X , then z ∈ Y if and only if there exists afilter base B consisting of subsets of Y which converges to z . The cardinality of the set S(Y ) of such filter bases isat most 22|Y | . Moreover, in a Hausdorff space, there is at most one limit to every filter base. Therefore, there is asurjection S(Y ) → X when Y = X.

The same arguments establish a more general result: suppose that a Hausdorff topological space X contains a densesubset of cardinality κ . Then X has cardinality at most 22κ and cardinality at most 2κ if it is first countable.The product of at most continuum many separable spaces is a separable space (Willard 1970, p. 109, Th 16.4c). Inparticular the space RR of all functions from the real line to itself, endowed with the product topology, is a separableHausdorff space of cardinality 2c . More generally, if κ is any infinite cardinal, then a product of at most 2κ spaceswith dense subsets of size at most κ has itself a dense subset of size at most κ (Hewitt–Marczewski–Pondiczerytheorem).

5.4 Constructive mathematics

Separability is especially important in numerical analysis and constructive mathematics, since many theorems that canbe proved for nonseparable spaces have constructive proofs only for separable spaces. Such constructive proofs canbe turned into algorithms for use in numerical analysis, and they are the only sorts of proofs acceptable in constructiveanalysis. A famous example of a theorem of this sort is the Hahn–Banach theorem.

5.5 Further examples

5.5.1 Separable spaces

• Every compact metric space (or metrizable space) is separable.

• The space C(K) of all continuous functions from a compact subsetK ⊆ R to the real line R is separable.

• The Lebesgue spacesLp (X,µ) , over a separable measure space ⟨X,M, µ⟩ , are separable for any 1 ≤ p < ∞.

• Any topological space which is the union of a countable number of separable subspaces is separable. Together,these first two examples give a different proof that n -dimensional Euclidean space is separable.

• The space C([0, 1]) of continuous real-valued functions on the unit interval [0, 1] with the metric of uniformconvergence is a separable space, since it follows from the Weierstrass approximation theorem that the setQ[x] of polynomials in one variable with rational coefficients is a countable dense subset of C([0, 1]) . TheBanach-Mazur theorem asserts that any separable Banach space is isometrically isomorphic to a closed linearsubspace of C([0, 1]) .

• A Hilbert space is separable if and only if it has a countable orthonormal basis. It follows that any separable,infinite-dimensional Hilbert space is isometric to the space ℓ2 of square-summable sequences.

• An example of a separable space that is not second-countable is the Sorgenfrey line S , the set of real numbersequipped with the lower limit topology.

24 CHAPTER 5. SEPARABLE SPACE

5.5.2 Non-separable spaces

• The first uncountable ordinal ω1 , equipped with its natural order topology, is not separable.

• The Banach space ℓ∞ of all bounded real sequences, with the supremum norm, is not separable. The sameholds for L∞ .

• The Banach space of functions of bounded variation is not separable; note however that this space has veryimportant applications in mathematics, physics and engineering.

5.6 Properties• A subspace of a separable space need not be separable (see the Sorgenfrey plane and the Moore plane), butevery open subspace of a separable space is separable, (Willard 1970, Th 16.4b). Also every subspace of aseparable metric space is separable.

• In fact, every topological space is a subspace of a separable space of the same cardinality. A constructionadding at most countably many points is given in (Sierpinski 1952, p. 49); if the space was a Hausdorff spacethen the space constructed which it embeds into is also a Hausdorff space.

• The set of all real-valued continuous functions on a separable space has a cardinality less than or equal to c.This follows since such functions are determined by their values on dense subsets.

• From the above property, one can deduce the following: If X is a separable space having an uncountable closeddiscrete subspace, then X cannot be normal. This shows that the Sorgenfrey plane is not normal.

• For a compact Hausdorff space X, the following are equivalent:

(i) X is second countable.(ii) The space C(X,R) of continuous real-valued functions on X with the supremum norm isseparable.(iii) X is metrizable.

5.6.1 Embedding separable metric spaces

• Every separable metric space is homeomorphic to a subset of the Hilbert cube. This is established in the proofof the Urysohn metrization theorem.

• Every separable metric space is isometric to a subset of the (non-separable) Banach space l∞ of all boundedreal sequences with the supremum norm; this is known as the Fréchet embedding. (Heinonen 2003)

• Every separable metric space is isometric to a subset of C([0,1]), the separable Banach space of continuousfunctions [0,1]→R, with the supremum norm. This is due to Stefan Banach. (Heinonen 2003)

• Every separable metric space is isometric to a subset of the Urysohn universal space.

For nonseparable spaces:

• A metric space of density equal to an infinite cardinal α is isometric to a subspace of C([0,1]α, R), the spaceof real continuous functions on the product of α copies of the unit interval. (Kleiber 1969)

5.7 References• Heinonen, Juha (January 2003), Geometric embeddings of metric spaces (PDF), retrieved 6 February 2009

• Kelley, John L. (1975), General Topology, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90125-1,MR 0370454

5.7. REFERENCES 25

• Kleiber, Martin; Pervin, William J. (1969), “A generalized Banach-Mazur theorem”, Bull. Austral. Math. Soc.1: 169–173, doi:10.1017/S0004972700041411

• Sierpiński, Wacław (1952), General topology, Mathematical Expositions, No. 7, Toronto, Ont.: University ofToronto Press, MR 0050870

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

• Willard, Stephen (1970), General Topology, Addison-Wesley, ISBN 978-0-201-08707-9, MR 0264581

Chapter 6

Separated sets

In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space thatare related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of whentwo sets are separated or not is important both to the notion of connected spaces (and their connected components)as well as to the separation axioms for topological spaces.Separated sets should not be confused with separated spaces (defined below), which are somewhat related but differ-ent. Separable spaces are again a completely different topological concept.

6.1 Definitions

There are various ways in which two subsets of a topological space X can be considered to be separated.

• A and B are disjoint if their intersection is the empty set. This property has nothing to do with topology assuch, but only set theory; we include it here because it is the weakest in the sequence of different notions. Formore on disjointness in general, see: disjoint sets.

• A and B are separated in X if each is disjoint from the other’s closure. The closures themselves do not haveto be disjoint from each other; for example, the intervals [0,1) and (1,2] are separated in the real line R, eventhough the point 1 belongs to both of their closures. More generally in anymetric space, two open balls Br(x1) ={y:d(x1,y)<r} and Bs(x2) = {y:d(x2,y)<s} are separated whenever d(x1,x2) ≥ r+s. Note that any two separatedsets automatically must be disjoint.

• A and B are separated by neighbourhoods if there are neighbourhoods U of A and V of B such that U and Vare disjoint. (Sometimes you will see the requirement that U and V be open neighbourhoods, but this makesno difference in the end.) For the example of A = [0,1) and B = (1,2], you could take U = (−1,1) and V =(1,3). Note that if any two sets are separated by neighbourhoods, then certainly they are separated. If A andB are open and disjoint, then they must be separated by neighbourhoods; just take U := A and V := B. For thisreason, separatedness is often used with closed sets (as in the normal separation axiom).

• A and B are separated by closed neighbourhoods if there is a closed neighbourhood U of A and a closedneighbourhoodV ofB such thatU andV are disjoint. Our examples, [0,1) and (1,2], are not separated by closedneighbourhoods. You could make eitherU or V closed by including the point 1 in it, but you cannot make themboth closed while keeping them disjoint. Note that if any two sets are separated by closed neighbourhoods,then certainly they are separated by neighbourhoods.

• A and B are separated by a function if there exists a continuous function f from the space X to the real lineR such that f(A) = {0} and f(B) = {1}. (Sometimes you will see the unit interval [0,1] used in place of R inthis definition, but it makes no difference in the end.) In our example, [0,1) and (1,2] are not separated by afunction, because there is no way to continuously define f at the point 1. Note that if any two sets are separatedby a function, then they are also separated by closed neighbourhoods; the neighbourhoods can be given in termsof the preimage of f as U := f−1[-e,e] and V := f−1[1-e,1+e], as long as e is a positive real number less than1/2.

26

6.2. RELATION TO SEPARATION AXIOMS AND SEPARATED SPACES 27

• A and B are precisely separated by a function if there exists a continuous function f from X to R such thatf−1(0) = A and f−1(1) = B. (Again, you may also see the unit interval in place of R, and again it makes nodifference.) Note that if any two sets are precisely separated by a function, then certainly they are separatedby a function. Since {0} and {1} are closed in R, only closed sets are capable of being precisely separatedby a function; but just because two sets are closed and separated by a function does not mean that they areautomatically precisely separated by a function (even a different function).

6.2 Relation to separation axioms and separated spaces

The separation axioms are various conditions that are sometimes imposed upon topological spaces which can bedescribed in terms of the various types of separated sets. As an example, we will define the T2 axiom, which is thecondition imposed on separated spaces. Specifically, a topological space is separated if, given any two distinct pointsx and y, the singleton sets {x} and {y} are separated by neighbourhoods.Separated spaces are also called Hausdorff spaces or T2 spaces. Further discussion of separated spaces may be foundin the article Hausdorff space. General discussion of the various separation axioms is in the article Separation axiom.

6.3 Relation to connected spaces

Given a topological space X, it is sometimes useful to consider whether it is possible for a subset A to be separatedfrom its complement. This is certainly true if A is either the empty set or the entire space X, but there may be otherpossibilities. A topological space X is connected if these are the only two possibilities. Conversely, if a nonemptysubset A is separated from its own complement, and if the only subset of A to share this property is the empty set,then A is an open-connected component of X. (In the degenerate case where X is itself the empty set {}, authoritiesdiffer on whether {} is connected and whether {} is an open-connected component of itself.)For more on connected spaces, see Connected space.

6.4 Relation to topologically distinguishable points

Given a topological space X, two points x and y are topologically distinguishable if there exists an open set that onepoint belongs to but the other point does not. If x and y are topologically distinguishable, then the singleton sets {x}and {y} must be disjoint. On the other hand, if the singletons {x} and {y} are separated, then the points x and ymust be topologically distinguishable. Thus for singletons, topological distinguishability is a condition in betweendisjointness and separatedness.For more about topologically distinguishable points, see Topological distinguishability.

6.5 Sources• Stephen Willard, General Topology, Addison-Wesley, 1970. Reprinted by Dover Publications, New York,2004. ISBN 0-486-43479-6 (Dover edition).

Chapter 7

Separation axiom

For the axiom of set theory, see Axiom schema of separation.In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds oftopological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. Theseare sometimes called Tychonoff separation axioms, after Andrey Tychonoff.The separation axioms are axioms only in the sense that, when defining the notion of topological space, one couldadd these conditions as extra axioms to get a more restricted notion of what a topological space is. The modernapproach is to fix once and for all the axiomatization of topological space and then speak of kinds of topologicalspaces. However, the term “separation axiom” has stuck. The separation axioms are denoted with the letter “T” afterthe German Trennungsaxiom, which means “separation axiom.”The precise meanings of the terms associated with the separation axioms has varied over time, as explained in Historyof the separation axioms. It is important to understand the authors’ definition of each condition mentioned to knowexactly what they mean, especially when reading older literature.

7.1 Preliminary definitions

Before we define the separation axioms themselves, we give concrete meaning to the concept of separated sets (andpoints) in topological spaces. (Separated sets are not the same as separated spaces, defined in the next section.)The separation axioms are about the use of topological means to distinguish disjoint sets and distinct points. It’s notenough for elements of a topological space to be distinct (that is, unequal); we may want them to be topologicallydistinguishable. Similarly, it’s not enough for subsets of a topological space to be disjoint; we may want them to beseparated (in any of various ways). The separation axioms all say, in one way or another, that points or sets that aredistinguishable or separated in some weak sense must also be distinguishable or separated in some stronger sense.Let X be a topological space. Then two points x and y in X are topologically distinguishable if they do not haveexactly the same neighbourhoods (or equivalently the same open neighbourhoods); that is, at least one of them has aneighbourhood that is not a neighbourhood of the other (or equivalently there is an open set that one point belongs tobut the other point does not).Two points x and y are separated if each of them has a neighbourhood that is not a neighbourhood of the other; thatis, neither belongs to the other’s closure. More generally, two subsets A and B of X are separated if each is disjointfrom the other’s closure. (The closures themselves do not have to be disjoint.) All of the remaining conditions forseparation of sets may also be applied to points (or to a point and a set) by using singleton sets. Points x and y willbe considered separated, by neighbourhoods, by closed neighbourhoods, by a continuous function, precisely by afunction, if and only if their singleton sets {x} and {y} are separated according to the corresponding criterion.Subsets A and B are separated by neighbourhoods if they have disjoint neighbourhoods. They are separated by closedneighbourhoods if they have disjoint closed neighbourhoods. They are separated by a continuous function if thereexists a continuous function f from the space X to the real lineR such that the image f(A) equals {0} and f(B) equals{1}. Finally, they are precisely separated by a continuous function if there exists a continuous function f from X to Rsuch that the preimage f−1({0}) equals A and f−1({1}) equals B.These conditions are given in order of increasing strength: Any two topologically distinguishable points must be

28

7.2. MAIN DEFINITIONS 29

distinct, and any two separated points must be topologically distinguishable. Any two separated sets must be disjoint,any two sets separated by neighbourhoods must be separated, and so on.For more on these conditions (including their use outside the separation axioms), see the articles Separated sets andTopological distinguishability.

7.2 Main definitions

These definitions all use essentially the preliminary definitions above.Many of these names have alternative meanings in some of mathematical literature, as explained on History of theseparation axioms; for example, the meanings of “normal” and “T4" are sometimes interchanged, similarly “regular”and “T3", etc. Many of the concepts also have several names; however, the one listed first is always least likely to beambiguous.Most of these axioms have alternative definitions with the samemeaning; the definitions given here fall into a consistentpattern that relates the various notions of separation defined in the previous section. Other possible definitions can befound in the individual articles.In all of the following definitions, X is again a topological space.

• X is T0, or Kolmogorov, if any two distinct points in X are topologically distinguishable. (It will be a commontheme among the separation axioms to have one version of an axiom that requires T0 and one version thatdoesn't.)

• X is R0, or symmetric, if any two topologically distinguishable points in X are separated.

• X is T1, or accessible or Fréchet, if any two distinct points in X are separated. Thus, X is T1 if and only if itis both T0 and R0. (Although you may say such things as “T1 space”, “Fréchet topology”, and “Suppose thatthe topological space X is Fréchet”, avoid saying “Fréchet space” in this context, since there is another entirelydifferent notion of Fréchet space in functional analysis.)

• X is R1, or preregular, if any two topologically distinguishable points in X are separated by neighbourhoods.Every R1 space is also R0.

• X is Hausdorff, or T2 or separated, if any two distinct points in X are separated by neighbourhoods. Thus, Xis Hausdorff if and only if it is both T0 and R1. Every Hausdorff space is also T1.

• X is T2½, or Urysohn, if any two distinct points in X are separated by closed neighbourhoods. Every T₂½space is also Hausdorff.

• X is completely Hausdorff, or completely T2, if any two distinct points in X are separated by a continuousfunction. Every completely Hausdorff space is also T₂½.

• X is regular if, given any point x and closed set F in X such that x does not belong to F, they are separated byneighbourhoods. (In fact, in a regular space, any such x and F will also be separated by closed neighbourhoods.)Every regular space is also R1.

• X is regular Hausdorff, or T3, if it is both T0 and regular.[1] Every regular Hausdorff space is also T₂½.

• X is completely regular if, given any point x and closed set F in X such that x does not belong to F, they areseparated by a continuous function. Every completely regular space is also regular.

• X is Tychonoff, or T3½, completely T3, or completely regular Hausdorff, if it is both T0 and completelyregular.[2] Every Tychonoff space is both regular Hausdorff and completely Hausdorff.

• X is normal if any two disjoint closed subsets of X are separated by neighbourhoods. (In fact, a space is normalif and only if any two disjoint closed sets can be separated by a continuous function; this is Urysohn’s lemma.)

30 CHAPTER 7. SEPARATION AXIOM

• X is normal Hausdorff, or T4, if it is both T1 and normal. Every normal Hausdorff space is both Tychonoffand normal regular.

• X is completely normal if any two separated sets are separated by neighbourhoods. Every completely normalspace is also normal.

• X is completely normal Hausdorff, or T5 or completely T4, if it is both completely normal and T1. Everycompletely normal Hausdorff space is also normal Hausdorff.

• X is perfectly normal if any two disjoint closed sets are precisely separated by a continuous function. Everyperfectly normal space is also completely normal.

• X is perfectly normal Hausdorff, or T6 or perfectly T4, if it is both perfectly normal and T1. Every perfectlynormal Hausdorff space is also completely normal Hausdorff.

7.3 Relationships between the axioms

The T0 axiom is special in that it can be not only added to a property (so that completely regular plus T0 is Tychonoff)but also subtracted from a property (so that Hausdorff minus T0 is R1), in a fairly precise sense; see Kolmogorovquotient for more information. When applied to the separation axioms, this leads to the relationships in the tablebelow:In this table, you go from the right side to the left side by adding the requirement of T0, and you go from the left sideto the right side by removing that requirement, using the Kolmogorov quotient operation. (The names in parenthesesgiven on the left side of this table are generally ambiguous or at least less well known; but they are used in the diagrambelow.)Other than the inclusion or exclusion of T0, the relationships between the separation axioms are indicated in thefollowing diagram:In this diagram, the non-T0 version of a condition is on the left side of the slash, and the T0 version is on the rightside. Letters are used for abbreviation as follows: “P” = “perfectly”, “C” = “completely”, “N” = “normal”, and “R”(without a subscript) = “regular”. A bullet indicates that there is no special name for a space at that spot. The dash atthe bottom indicates no condition.You can combine two properties using this diagram by following the diagram upwards until both branches meet.For example, if a space is both completely normal (“CN”) and completely Hausdorff (“CT2"), then following bothbranches up, you find the spot "•/T5". Since completely Hausdorff spaces are T0 (even though completely normalspaces may not be), you take the T0 side of the slash, so a completely normal completely Hausdorff space is the sameas a T5 space (less ambiguously known as a completely normal Hausdorff space, as you can see in the table above).As you can see from the diagram, normal and R0 together imply a host of other properties, since combining the twoproperties leads you to follow a path through the many nodes on the rightside branch. Since regularity is the mostwell known of these, spaces that are both normal and R0 are typically called “normal regular spaces”. In a somewhatsimilar fashion, spaces that are both normal and T1 are often called “normal Hausdorff spaces” by people that wishto avoid the ambiguous “T” notation. These conventions can be generalised to other regular spaces and Hausdorffspaces.

7.4 Other separation axioms

There are some other conditions on topological spaces that are sometimes classified with the separation axioms, butthese don't fit in with the usual separation axioms as completely. Other than their definitions, they aren't discussedhere; see their individual articles.

• X is semiregular if the regular open sets form a base for the open sets of X. Any regular space must also besemiregular.

7.5. SEE ALSO 31

• X is quasi-regular if for any nonempty open set G, there is a nonempty open set H such that the closure of His contained in G.

• X is fully normal if every open cover has an open star refinement. X is fully T4, or fully normal Hausdorff,if it is both T1 and fully normal. Every fully normal space is normal and every fully T4 space is T4. Moreover,one can show that every fully T4 space is paracompact. In fact, fully normal spaces actually have more to dowith paracompactness than with the usual separation axioms.

• X is sober if, for every closed set C that is not the (possibly nondisjoint) union of two smaller closed sets, thereis a unique point p such that the closure of {p} equals C. More briefly, every irreducible closed set has a uniquegeneric point. Any Hausdorff space must be sober, and any sober space must be T0.

7.5 See also• General topology

7.6 Sources• Schechter, Eric (1997). Handbook of Analysis and its Foundations. San Diego: Academic Press. ISBN0126227608. (has Ri axioms, among others)

• Willard, Stephen (1970). General topology. Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0-486-43479-6.(has all of the non-Ri axioms mentioned in the Main Definitions, with these definitions)

• Merrifield, Richard E.; Simmons, Howard E. (1989). Topological Methods in Chemistry. New York: Wiley.ISBN 0-471-83817-9. (gives a readable introduction to the separation axioms with an emphasis on finitespaces)

[1] Schechter, p. 441

[2] Schechter, p. 443

7.7 External links• Separation Axioms at ProvenMath

• Table of separation and metrisability axioms from Schechter

32 CHAPTER 7. SEPARATION AXIOM

An illustration of some of the separation axioms. A blue region indicates an open set, a red rectangle a closed set, and a black dot apoint.

7.7. EXTERNAL LINKS 33

Hasse diagram of the separation axioms.

Chapter 8

Topological indistinguishability

In topology, two points of a topological space X are topologically indistinguishable if they have exactly the sameneighborhoods. That is, if x and y are points in X, and N_x is the set of all neighborhoods that contain x, and N_yis the set of all neighborhoods that contain y, then x and y are “topologically indistinguishable” if and only if N_x =N_y. (See Hausdorff’s axiomatic neighborhood systems.)Intuitively, two points are topologically indistinguishable if the topology of X is unable to discern between the points.Two points of X are topologically distinguishable if they are not topologically indistinguishable. This means thereis an open set containing precisely one of the two points (equivalently, there is a closed set containing precisely oneof the two points). This open set can then be used to distinguish between the two points. A T0 space is a topologicalspace in which every pair of distinct points is topologically distinguishable. This is the weakest of the separationaxioms.Topological indistinguishability defines an equivalence relation on any topological space X. If x and y are points of Xwe write x ≡ y for "x and y are topologically indistinguishable”. The equivalence class of x will be denoted by [x].

8.1 Examples

For T0 spaces (in particular, for Hausdorff spaces) the notion of topological indistinguishability is trivial, so one mustlook to non-T0 spaces to find interesting examples. On the other hand, regularity and normality do not imply T0, sowe can find examples with these properties. In fact, almost all of the examples given below are completely regular.

• In an indiscrete space, any two points are topologically indistinguishable.

• In a pseudometric space, two points are topologically indistinguishable if and only if the distance between themis zero.

• In a seminormed vector space, x ≡ y if and only if ‖x − y‖ = 0.

• For example, let L2(R) be the space of all measurable functions from R to R which are square integrable(see Lp space). Then two functions f and g in L2(R) are topologically indistinguishable if and only if theyare equal almost everywhere.

• In a topological group, x ≡ y if and only if x−1y ∈ cl{e} where cl{e} is the closure of the trivial subgroup. Theequivalence classes are just the cosets of cl{e} (which is always a normal subgroup).

• Uniform spaces generalize both pseudometric spaces and topological groups. In a uniform space, x ≡ y ifand only if the pair (x, y) belongs to every entourage. The intersection of all the entourages is an equivalencerelation on X which is just that of topological indistinguishability.

• Let X have the initial topology with respect to a family of functions {fα : X → Yα} . Then two points x andy in X will be topologically indistinguishable if the family fα does not separate them (i.e. fα(x) = fα(y) forall α ).

34

8.2. SPECIALIZATION PREORDER 35

• Given any equivalence relation on a set X there is a topology on X for which the notion of topological indistin-guishability agrees with the given equivalence relation. One can simply take the equivalence classes as a basefor the topology. This is called the partition topology on X.

8.2 Specialization preorder

The topological indistinguishability relation on a space X can be recovered from a natural preorder on X called thespecialization preorder. For points x and y in X this preorder is defined by

x ≤ y if and only if x ∈ cl{y}

where cl{y} denotes the closure of {y}. Equivalently, x ≤ y if the neighborhood system of x, denoted Nx, is containedin the neighborhood system of y:

x ≤ y if and only if Nx ⊂ Ny.

It is easy to see that this relation on X is reflexive and transitive and so defines a preorder. In general, however, thispreorder will not be antisymmetric. Indeed, the equivalence relation determined by ≤ is precisely that of topologicalindistinguishability:

x ≡ y if and only if x ≤ y and y ≤ x.

A topological space is said to be symmetric (or R0) if the specialization preorder is symmetric (i.e. x ≤ y implies y ≤x). In this case, the relations ≤ and ≡ are identical. Topological indistinguishability is better behaved in these spacesand easier to understand. Note that this class of spaces includes all regular and completely regular spaces.

8.3 Properties

8.3.1 Equivalent conditions

There are several equivalent ways of determining when two points are topologically indistinguishable. Let X be atopological space and let x and y be points of X. Denote the respective closures of x and y by cl{x} and cl{y}, andthe respective neighborhood systems by Nx and Ny. Then the following statements are equivalent:

• x ≡ y

• for each open set U in X, either U contains both x and y or neither of them

• Nx = Ny

• x ∈ cl{y} and y ∈ cl{x}

• cl{x} = cl{y}

• x ∈ ∩Ny and y ∈ ∩Nx

• ∩Nx = ∩Ny

• x ∈ cl{y} and x ∈ ∩Ny

• x belongs to every open set and every closed set containing y

• a net or filter converges to x if and only if it converges to y

These conditions can be simplified in the case where X is symmetric space. For these spaces (in particular, for regularspaces), the following statements are equivalent:

36 CHAPTER 8. TOPOLOGICAL INDISTINGUISHABILITY

• x ≡ y

• for each open set U, if x ∈ U then y ∈ U

• Nx ⊂ Ny

• x ∈ cl{y}

• x ∈ ∩Ny

• x belongs to every closed set containing y

• x belongs to every open set containing y

• every net or filter that converges to x converges to y

8.3.2 Equivalence classes

To discuss the equivalence class of x, it is convenient to first define the upper and lower sets of x. These are bothdefined with respect to the specialization preorder discussed above.The lower set of x is just the closure of {x}:

↓x = {y ∈ X : y ≤ x} = cl{x}

while the upper set of x is the intersection of the neighborhood system at x:

↑x = {y ∈ X : x ≤ y} =∩

Nx.

The equivalence class of x is then given by the intersection

[x] = ↓x ∩ ↑x.

Since ↓x is the intersection of all the closed sets containing x and ↑x is the intersection of all the open sets containingx, the equivalence class [x] is the intersection of all the open and closed sets containing x.Both cl{x} and ∩Nx will contain the equivalence class [x]. In general, both sets will contain additional points as well.In symmetric spaces (in particular, in regular spaces) however, the three sets coincide:

[x] = cl{x} =∩

Nx.

In general, the equivalence classes [x] will be closed if and only if the space is symmetric.

8.3.3 Continuous functions

Let f : X→ Y be a continuous function. Then for any x and y in X

x ≡ y implies f(x) ≡ f(y).

The converse is generally false (There are quotients of T0 spaces which are trivial). The converse will hold ifX has theinitial topology induced by f. More generally, if X has the initial topology induced by a family of maps fα : X → Yα

then

x ≡ y if and only if fα(x) ≡ fα(y) for all α.

It follows that two elements in a product space are topologically indistinguishable if and only if each of their compo-nents are topologically indistinguishable.

8.4. KOLMOGOROV QUOTIENT 37

8.4 Kolmogorov quotient

Since topological indistinguishability is an equivalence relation on any topological space X, we can form the quotientspace KX = X/≡. The space KX is called the Kolmogorov quotient or T0 identification of X. The space KX is, infact, T0 (i.e. all points are topologically distinguishable). Moreover, by the characteristic property of the quotientmap any continuous map f : X→ Y from X to a T0 space factors through the quotient map q : X→ KX.Although the quotient map q is generally not a homeomorphism (since it is not generally injective), it does induce abijection between the topology on X and the topology on KX. Intuitively, the Kolmogorov quotient does not alter thetopology of a space. It just reduces the point set until points become topologically distinguishable.

8.5 See also• T0 space

• Specialization preorder

38 CHAPTER 8. TOPOLOGICAL INDISTINGUISHABILITY

8.6 Text and image sources, contributors, and licenses

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