LECTURE NOTES FOR GENERAL TOPOLOGY

BIT, SPRING 2018

DAVID G.L. WANG†‡

Contents

1. Introduction 2

1.1. Who cares topology? 2

1.2. Geometry v.s. topology 4

1.3. Applications of topology 5

1.4. The origin of topology 5

1.5. Topological equivalence 7

1.6. Surfaces 8

1.7. Abstract spaces 8

1.8. The classification theorem and more 9

Recap 11

2. Notion and notation 12

3. Continuity 15

3.1. Open and closed sets 15

3.2. Continuous functions 22

Date: March 7, 2018.

2 D.G.L. WANG

1. Introduction

1.1. Who cares topology?

Question 1.1. Imagine you have super stretchy and bendy pants. Can you turnthem inside out without taking your feet off the ground?

Answer. Yes.

Question 1.2. You have a picture frame with a loop of string fixed on the back,and two nails. Can you hang it onto the wall using both nails so that pulling outany nail from the wall leads the painting falling down?

Figure 1.1. A solution to Question 1.2. Screenshot from a video ofthe PBS Digital Studios.

Answer. Yes.

Question 1.3. The necklace splitting problem. Its name and solutions are due tomathematicians Alon & West. See Fig. 1.2.

Figure 1.2. Illustration of one version of the necklace splittingproblem. Screenshot from a video of 3Blue1Brown.

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Answer. The version illustrated in Fig. 1.2 can be solved by using Borsuk-Ulamtheorem: every continuous function from an n-sphere into En maps some pair ofantipodal points to the same point.

Question 1.4. The inscribed square problem: Any simple closed curve inscribes asquare. See Fig. 1.3.

Figure 1.3. A simple, closed, polygonal curve inscribing a square.Stolen from Terrence Tao’s manuscript on arXiv:1611.07441.

Answer. Open up to 2017. The proposition that any simple closed curve inscribe arectangle can be shown with the aid of Mobius strips.

Figure 1.4. August Ferdinand Mobius (1790–1868) was a Germanmathematician and theoretical astronomer. The right part is aMobius strip. Stolen from Wiki.

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Homework 1.1. Make a Mobius strip by yourself.

The Nobel Prize in Physics 2016 was awarded with one half to David J. Thouless,and the other half to F. Duncan M. Haldane and J. Michael Kosterlitz “for theo-retical discoveries of topological phase transitions and topological phasesof matter”.

Figure 1.5. Topology was the key to the Nobel Laureates’ discov-eries, and it explains why electronical conductivity inside thin layerschanges in integer steps. Stolen from Popular Science Background ofthe Nobel Prize in Physics 2016, Page 4(5).

1.2. Geometry v.s. topology. Below are some views from Robert MacPherson,a plenary addresser at the ICM in Warsaw in 1983.

• Geometry (from ancient Greek): geo=earth, metry=measurement.Topology (from Greek): τ oπoσ=place/position, λoγoσ=study/discourse.

• Topology is “geometry” without without measurement.It is qualitative (as opposed to quantitative) “geometry”.

• Geometry: The point M is the midpoint of the straight line segment L connectingA to B.Topology: The point M lies on the curve L connecting A to B.

• Geometry calls its objects configurations (circles, triangles, etc.)Topology calls its objects spaces.

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1.3. Applications of topology. Topology, over most of its history, has NOTgenerally been applied outside of mathematics (with a few interesting exceptions).

WHY?

• TOO abstract? The ancient mathematicians could not even convince of the subject.

Figure 1.6. Screenshot from a video of Robert MacPherson’s talkin Institute for Advanced Study.

• It is qualitative, not quantitative? People think of science as a quantitativeendeavour.

1.4. The origin of topology. Three stories.

1.4.1. The seven bridges of Konigsberg. The problem was to devise a walk throughthe city that would cross each of those bridges once and only once.

The negative resolution by Leonhard Paul Euler (1707–1783) in 1736 laid thefoundations of graph theory and prefigured the idea of topology. The difficulty Eulerfaced was the development of a suitable technique of analysis, and of subsequenttests that established this assertion with mathematical rigor.

Euler was a Swiss mathematician, physicist, astronomer, logician and engineerwho made important and influential discoveries in many branches of mathematicslike infinitesimal calculus and graph theory, while also making pioneering contribu-tions to several branches such as topology and analytic number theory.

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Figure 1.7. Map of Konigsberg in Euler’s time showing the actuallayout of the seven bridges, highlighting the river Pregel and thebridges. Stolen from Wiki.

1.4.2. The four colour theorem. The four colour theorem states that given anyseparation of a plane into contiguous regions, producing a figure called a map, nomore than four colours are required to colour the regions of the map so that notwo adjacent regions have the same color.

Figure 1.8. A four-coloring of a map. Stolen from Google image.

The four color theorem was proved in 1976 by Kenneth Appel and WolfgangHaken. It was the first major theorem to be proved using a computer.

1.4.3. Euler characteristic. χ = v − e+ f .

We use the terminology polyhedron to indicate a surface rather than a solid.

Theorem 1.5 (Euler’s polyhedral formula). Let P be a polyhedron s.t.

• Any two vertices of P can be connected by a chain of edges.

• Any cycle along edges of P which is made up of straight line segments (notnecessarily edges) separates P into 2 pieces.

Then the Euler number or Euler characteristic χ = 2 for P .

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Figure 1.9. Stamp of the former German Democratic Republichonouring Euler on the 200th anniversary of his death. Across thecentre it shows his polyhedral formula. Stolen from Wiki.

• 1750: first appear in a letter from Euler to Christian Goldbach (1690–1764).

• 1860 (around): Mobius gave the idea of explaining topological equivalence bythinking of spaces as being made of rubber. It works for concave P .

• David Eppstein collected 20 proofs of Theorem 1.5.

Homework 1.2. Compute the Euler characteristic of the Platonic solids (allregular and convex polyhedra): the tetrahedron, the cube, the octahedron, thedodecahedron, and the icosahedron.

Figure 1.10. The Plotonic solids. Stolen from Wiki.

1.5. Topological equivalence. = homeomorphism; see Section 1.7.

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√: thicken, stretch, bend, twist, . . .;

× : identify, tear, . . ..

Figure 1.11. Some surfaces which are not equivalent. Stolen fromHaldane’s slides on Dec. 8, 2016.

Theorem 1.6. Topological equivalent polyhedra have the same Euler characteristic.

• The starting point for modern topology.

• Search for unchanged properties of spaces under topological equivalence.

• χ = 2 belongs to S2, rather than to particular polyhedra → define χ for S2.

• Theorem 1.6: different calculations, same answer.

1.6. Surfaces. What exactly do we mean by a “space”?

• Homeomorphism → Continuity.

• Geometry → Boundedness.

1.7. Abstract spaces. The axioms for a topological space appearing for the firsttime in 1914 in the work of Felix Hausdorff (1868–1942). Hausdorff, a Germanmathematician, is considered to be one of the founders of modern topology, whocontributed significantly to set theory, descriptive set theory, measure theory,function theory, and functional analysis.

How has modern definition of a topological space been formed?

(1) General enough to allow set of points or functions, and performable construc-tions like the Cartesian products and the identifying. Enough information todefine the continuity of functions between spaces.

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(2) Cauchy: distances → continuity.

(3) No distance! Continuity ← neighbourhood ← axiom.

A function f : Em → En is continuous if given any x ∈ Em and any neighbour-hood N of f(x), then f−1(N) is a neighbourhood of x.

Definition 1.7. Let X be a set. Suppose that for any x ∈ X, a collection Nx ∈ 22X

is assigned to x, satisfying the following 4 axioms:

(a) If x ∈ X and N ∈ Nx, then x ∈ N .

(b) If x ∈ X and N1, N2 ∈ Nx, then N1 ∩N2 ∈ Nx.(c) If x ∈ X, N ∈ Nx, and N ⊂M , then M ∈ Nx.(d) If x ∈ X and N ∈ Nx, then y ∈ N : N ∈ Ny ∈ Nx.

Then we call every subset N ∈ Nx a neighbourhood of x. The assignment(x,Nx) : x ∈ X of the collection of neighbourhoods satisfying axioms (a)–(d) toeach x ∈ X is called a topology on X. The whole structure, that is, the set Xtogether with the topology, is called a topological space. A function f : X → Ybetween topological spaces is continuous if ∀x ∈ X and ∀N ∈ Nf(x), the setf−1(N) ∈ Nx. It is called a homeomorphism if it is a continuous bijection andhas a continuous inverse. We call X and Y are homeomorphic or topologicallyequivalent spaces if such a function exists.

• The Euclidean space En.

• The subspace topology → surfaces become topological spaces.

• Metric → topology; e.g., d(f, g) = supx |f(x)− g(x)|.• The peculiar finite-complement topology: N ⊆ R containing x ∈ R is a

neighbourhood of x ⇐⇒ |R \N | <∞.

• E.g.: x ∈ Nx, ∀x ∈ X =⇒ ∀ f : X → Y is continuous.

1.8. The classification theorem and more.

Theorem 1.8 (Classification theorem). Any closed surface is homeomorphic to S2

with either a finite number of handles added, or a finite number of Mobius stripsadded. No two of these surfaces are homeomorphic.

Definition 1.9. The S2 with n handles added is called an orientable surface ofgenus n. Non-orientable surfaces can be defined analogously.

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Historical notes. The classification of surfaces was initiated and carried throughin the orientable case by Mobius in a paper which he submitted for consideration forthe Grand Prix de Mathematiques of the Paris Academy of Sciences. He was 71 atthe time. The jury did not consider any of the manuscripts received as being worthyof the prize, and Mobius’ work finally appeared as just another mathematical paper.

Decide ∼= or 6∼=.

• ∼=: construct a homeomorphism; techniques vary.

• 6∼=: look for topological invariants, e.g., geometric properties, numbers, algebraicsystems.

Examples to show 6∼=.

• E1 6∼= E2: connectedness, h : E1 \ 0 → E2 \ h(0).• Poincare’s construction idea: assign a group to each topological space so that

homeomorphic spaces have isomorphic groups. But group isomorphism does notimply homeomorphism.

Here are some theorems that the fundamental groups help prove.

Theorem 1.10 (Classification of surfaces). No 2 surfaces in Theorem 1.8 haveisomorphic fundamental groups.

Theorem 1.11 (Jordan separation theorem). Any simple closed curve in E1

divides E1 into 2 pieces.

Figure 1.12. Marie Ennemond Camille Jordan (1838–1922) wasa French mathematician, known both for his foundational work ingroup theory and for his influential Cours d’analyse. The Jordancurve (drawn in black) divides the plane into an “inside” region (lightblue) and an “outside” region (pink). Stolen from Wiki.

Theorem 1.12 (Brouwer fixed-point theorem). Any continuous function from adisc to itself leaves at least one point fixed.

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Figure 1.13. Luitzen Egbertus Jan Brouwer (1881–1966), usuallycited as L. E. J. Brouwer but known to his friends as Bertus, was aDutch mathematician and philosopher, who worked in topology, settheory, measure theory and complex analysis. He was the founder ofthe mathematical philosophy of intuitionism. Stolen from Wiki.

Theorem 1.13 (Nielsen-Schreier theorem). A subgroup of a free group is alwaysfree.

Recap

Please ask yourself the following questions.

(1) How old is topology?

(2) Did you check that what the Borsuk-Ulam theorem is all about?

(3) Have you ever made a Mobius strip by yourself? What happens if you cut italong the middle line that is parallel to its boundary?

(4) Why Euler’s answer the seven bridges problem is negative?

(5) What is the definition of a Platonic solid? How many Platonic solids are there?Have you ever tried to compute the Euler’s characteristic of the icosahedron anddodecohedron?

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2. Notion and notation

List 1.

1) set, collection, family, class.

2) element, member.

3) the empty set: ∅.4) singleton: x.5) nonnegative integers N; integers Z; rational numbers Q; real numbers R; complex

numbers C.

6) list notation v.s. set-builder notation:

−1, −2, −3, . . . = x ∈ Z |x < 0 = x ∈ Z : x < 0.

7) x ∈ A: the element x belongs to the set A.

8) A ⊆ B: the set A is a subset of the set B, or B includes A.

A ⊂ B: A ⊆ B and A 6= B.

9) Criterion of Equality for sets:

A = B ⇐⇒ A ⊆ B & B ⊆ A.

10) X \ A: the difference of the sets X and A.

If A ⊆ X, then X \ A is called the complement of A in X, denoted by Ac ifthe underlying set X is known well without confusion.

List 2.

1) union, join: A ∪B.

2) intersection, meet: A ∩B.

3) disjoint: A ∩B = ∅.4) Venn diagrams v.s. Euler diagrams.

5) ∪ and ∩ satisfy the following laws.

(a) commutative laws: A ∪B = B ∪ A and A ∩B = B ∩ A.

(b) associated laws:

(A ∪B) ∪ C = A ∪ (B ∪ C) and (A ∩B) ∩ C = A ∩ (B ∩ C).

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(c) distributive laws:

(A ∪B) ∩ C = (A ∩ C) ∪ (B ∩ C) and (A ∩B) ∪ C = (A ∪ C) ∩ (B ∪ C).

(d) de Morgan’s laws:

• the complement of the union is the intersection of the complements:(⋃A∈Λ

A

)c=⋂A∈Λ

(Ac).

• the complement of an intersection is the union of the complements:(⋂A∈Λ

A

)c=⋃A∈Λ

(Ac).

6) symmetric difference of A and B:

A∆B = (A \B) ∪ (B \ A) = (A ∪B) \ (A ∩B).

Homework 2.1. Show the associativity of symmetric difference

(A∆B) ∆C = A∆ (B∆C).

Note that the associativity makes the expression A∆B∆C meaningful.

Due to a mistake in an old version of this note, Homework 2.1 and Homework 2.2are not considered to evaluate your final score. — Mar. 4th, 2018.

Homework 2.2. Prove or disprove the following formulas.

i) (A∆B) ∪ C = (A∆C) ∪ (B∆C).

ii) (A∆B) ∩ C = (A∆C) ∩ (B∆C).

Answer. By using the Venn’s diagram. A correct distributive law for the symmetricdifference is

(A∆B) ∩ C = (A ∩ C) ∆ (B ∩ C).

List 3.

1) supremum, denoted sup.

2) infimum, denoted inf.

3) The Greek alphabet:

α, β, γ, δ, ε(ε), ζ, η, θ(ϑ), ι, κ, λ, µ, ν, ξ,

o, π($), ρ(%), σ(ς), τ, υ, φ(ϕ), χ, ψ, ω.

A, B, Γ(Γ ), ∆(∆), E, Z, H, Θ(Θ), I, K, Λ(Λ), M, N, Ξ,

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O, Π, P, Σ, T, Υ(Υ ), Φ(Φ), X, Ψ(Ψ), Ω(Ω).

Comments:

• Euclid’s famous mathematical treatise is also called Elements.

• The empty set is everywhere.

• Inclusion is both reflexive and transitive;belonging is neither reflexive nor transitive.

• Unlike Venn diagrams, which show all possible relations between different sets,the Euler diagram shows only relevant relationships.

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3. Continuity

The definition of topological space fits quite well with our intuitive idea of whata space ought to be. Unfortunately it is not terribly convenient to work with. Wewant an equivalent, more manageable, set of axioms!

3.1. Open and closed sets. We define topological spaces by using open sets.

Definition 3.1. Let X be a set.

• A topology or topological structure on X: a collection of subsets of X,denoted Ω, satisfying the axioms

(i) ∅ ∈ Ω and X ∈ Ω;

(ii) ∪λOλ ∈ Ω, ∀ Oλλ ⊂ Ω;

(iii) O1 ∩O2 ∈ Ω, ∀O1, O2 ∈ Ω.

• A topological space: the pair (X,Ω).

• A point of (X,Ω): an element of X.

• An open set in (X,Ω): a member in Ω.

• A closed set in (X,Ω): a subset A ⊆ X s.t. Ac ∈ Ω.

• A clopen set in (X,Ω): a subset A ⊆ X which is both closed and open.

• A neighbourhood of a point p ∈ X: a subset N ⊆ X s.t. ∃ O ∈ Ω s.t.p ∈ O ⊆ N .

Remark 1. A set might be open but not closed, closed but not open, clopen, orneither closed nor open.

Remark 2. One may show that Def. 3.1 is compatible with Def. 1.7.

Remark 3. We are following N. Bourbaki that analysts and French mathematiciansdefine the term “neighbourhood” in the above sense. There is another custom thata neighbourhood of a point p is an open set containing p.

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Remark 4. Why do we use the letter O and the letter Ω?

• Open in English

• Ouvert in French

• Otkrytyj in Russian

• Offen in German

• Oppen in Swedish

• Otvoren in Croatian

• Otevreno in Czech

• Open in Dutch

Example 3.2. Some topological spaces go along with this course.

(1) The discrete topological space: (X, 2X).

(2) The indiscrete or trivial topological space (X, ∅, X).(3) The real line: (R, ΩR = the union of any open intervals).

ΩR: the canonical or standard topology on R.

Question 3.3. The set S = 0 ∪ 1/n : n ∈ Z+ is closed in the real line R.

Answer. The complement

R \ S = (−∞, 0) ∪ (1/(n+ 1), 1/n) : n ∈ Z+ ∪ (1, +∞)

is the union of open intervals, and thus open. From definition, the set S is closed.

“Think geometrically, prove algebraically.” — John Tate

Example 3.4. The Cantor ternary setK is the number set created by iterativelydeleting the open middle third from a set of line segments, i.e.,

K =

∑k≥1

ak3k

: ak ∈ 0, 2

=

0.a1a2 · · · : ai 6∈ 1, 4, 7⊂ [0, 1].

The set K is closed in R.

The Cantor ternary set was discovered in 1874 by Henry John Stephen Smith,and introduced by Georg Cantor (1845–1918) in 1883. The Cantor ternary set K

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Figure 3.1. John Tate (1925–) is an American mathematician, dis-tinguished for many fundamental contributions in algebraic numbertheory, arithmetic geometry and related areas in algebraic geometry.He is professor emeritus at Harvard University. He was awarded theAbel Prize in 2010. Stolen from Wiki.

has a number of remarkable and deep properties. For instance, the set K does notcontain any interval of non-zero length, since

1−∑n≥0

2n

3n+1= 0.

Figure 3.2. Georg Cantor (1845–1918) was a German mathemati-cian. He invented set theory. The right part illustrates the Cantorternary set. Stolen from Wiki and Math Counterexamples respec-tively.

Question 3.5. Can you restate the axioms of a topology structure in terms ofclosed sets? Is your statement robust enough to be considered as an alternativedefinition of a topology?

Answer. Here is a robust one.

(i)’ ∅ and X are closed;

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(ii)’ the union of any finite number of closed sets is closed;

(iii)’ the intersection of any collection of closed sets is closed.

Homework 3.1 (Deadline: Mar. 14th). Find a topological space (X,Ω) with aset A ⊂ X satisfying all the following properties.

a) A is neither open nor closed;

b) A is the union of an infinite number of closed sets; and

c) A is the intersection of an infinite number of open sets.

Answer. The interval [ 0, 1) in the real line R.

Example 3.6. Some other examples for topological spaces.

(1) The arrow space (X,Ω):

X = [ 0, ∞) and Ω = ∅, X ∪ (a,∞) : a ≥ 0.

(2) The finite-complement topology or the T1-topology:

ΩT1 = ∅, X ∪ complement of finite subsets of R.The line with T1-topology (R, ΩT1).

(3) Let p ∈ X.

• A particular point topology: Ω = ∅, X ∪ S ⊆ X : p ∈ S.• An excluded point topology: Ω = ∅, X ∪ S ⊆ X : p 6∈ S.

Question 3.7. Does there exist a topology which is both a particular point topologyand an excluded point topology?

Question 3.8. For each of the following pairs (X,Ω), prove or disprove that it isa topological space.

(1) X is the plane, Ω = ∅, X ∪ open disks centerd at the origin.(2) X = R and Ω = ∅, X ∪ infinite subsets of R.

Homework 3.2 (Deadline: Mar. 14th). Find a smallest topological space whichis neither discrete nor indiscrete.

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Definition 3.9. Given (X,Ω) with A ⊆ X.

• The subspace topology (or induced topology) of A:

ΩA = O ∩ A : O ∈ Ω.

• The topological subspace induced by A: (A, ΩA).

• A limit point of A: a point p ∈ X s.t.

N ∩ (A− p) 6= ∅, ∀ neighbourhood N of p.

• An isolated point of A: a point p ∈ A with a neighbourhood N s.t.

N ∩ (A− p) = ∅.

• The closure of A: the union of A and its limit points, denoted A.

• An adherent point of A: a point in A.

• The interior of A: A = ∪O∈Ω∩AO.

• The exterior of A: (Ac).

• The boundary of A: ∂A = A \ A.

• A is (everywhere) dense if A = X.

• A is nowhere dense if (Ac) is everywhere dense.

Homework 3.3 (Deadline: Mar. 14th). Is there a topological space (X, Ω) withA ⊂ Y ⊂ X s.t. the set A is clopen in Y but neither open nor closed in X?

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Proposition 3.10. Topological spaces have the following properties.

(1) If X has a subspace Y , and Y has a subspace Z, then Z is a subspace of X.

(2) Consider the union U = ∪λXλ. If U ⊆ W , and each Xλ and U have the inducedtopology from W , then ΩU = ∪λΩλ.

(3) U c ∈ ΩA ⇐⇒ U = C ∩ A with Cc ∈ ΩX .

(4) Let A ⊆ X ⊆ Y . If Ac ∈ ΩX and Xc ∈ ΩY , then Ac ∈ ΩY .

(5) A is closed ⇐⇒ A = A ⇐⇒ A contains all its limit points.

(6) The closure of a set A is the smallest closed set containing A, because it is theintersection of all closed sets containing A.

(7) If A ⊆ B, then A ⊆ B.

(8) If A ∈ ΩX , then A = A. Thus (A) = A.

(9) A ∪B = A ∪B and (A ∩B) = A ∩B.

(10) A ∩B ⊆ A ∩B and (A ∪B) ⊇ A ∪B.

(11) Every dense set meets every nonempty O ∈ ΩX .

(12) If A is dense and O ∈ ΩX , then O ⊆ A ∩O.

(13) ∂A = A ∩ Ac = (A ∪ (Ac))c.

(14) ∂(A) ⊆ ∂A.

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Definition 3.11. Given (X,Ω) with β ⊆ Ω.

• β is a base for (X,Ω): ∀O ∈ Ω is the union of some members in β, i.e.,

β is a base ⇐⇒ ∀ x ∈ X, ∀ N ∈ Nx, ∃ B ∈ β s.t. x ∈ B ⊆ N.

• Ω is generated by β.

• A basic open set w.r.t. β: a member B ∈ β.

Example 3.12. All open intervals form a base for the real line.

Question 3.13. Here are some questions about bases.

(1) Is it possible that two distinct topologies have the same base?

(2) Describe a smallest base for a discrete space. Is it unique?

(3) Describe a smallest base for an indiscrete space. Is it unique?

(4) Describe a smallest base for the arrow space. Is it unique?

Remark 5. The bases of topological spaces have the following properties.

(1) In contrast to a basis of a vector space in linear algebra, a base need not to bemaximal, e.g., any open set can be safely added to a base.

(2) A topological space may have disjoint bases of distinct sizes, e.g., ΩE has a baseof all open intervals with rational ends, and another base of all open intervalswith irrational ends.

Definition 3.14. Given (X,Ω).

• X is second-countable if Ω has a countable base.

• X is separable if X contains a countable dense subset.

Any second-countable space is separable.

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3.2. Continuous functions.

Definition 3.15. Given (X,ΩX) and (Y,ΩY ). Let f : X → Y be a function.

• f is open: open 7→ open, i.e., f(ΩX) ⊆ ΩY .

• f is closed: closed 7→ closed.

• f is continuous if one of the following holds:

(1) f−1 is open, namely f−1(ΩY ) ⊆ ΩX .

(2) f−1 is closed.

(3) f−1(N) ∈ ΩX , ∀ neighbourhood N ⊆ Y .

(4) β is a base for ΩY , B ∈ β ⇒ f−1(B) ∈ ΩX .

(5) f(A) ⊆ f(A), ∀A ⊆ X.

(6) f−1(B) ⊆ f−1(B), ∀B ⊆ Y .

Definition 3.16. The inclusion or canonical injection ι : A→ X is defined byι(x) = x. Thus inclusions are always continuous, and called inclusion maps.

Proposition 3.17. Continuous functions have the following properties.

(1) Compositions, projections, and inclusions preserve continuity.

(2) The set of fixed points of any continuous f : R→ R is closed.

(3) The kernel of any continuous f : X → R is closed.

(4) Let f : E1 → E1 be continuous. Then its graph Γf : E1 → E2 defined by Γ(x) =(x, f(x)) is continuous, and Γf (E1) ∼= E1.

(5) Let X = ∪n≥1An where An ⊆ An+1 for each n. Let f : X → Y . If f |An iscontinuous for each n, then so is f .

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Definition 3.18. Given f : X → Y .

• f is a homeomorphism: f is a continuous open bijection, i.e., a continuousclosed bijection.

• X is homeomorphic to Y if ∃ a homeomorphism between them.

• f is an embedding: if f is a continuous injection s.t. f : X → f(X) is ahomeomorphism, denoted usually by using a hook right arrow as f : X → Y .

Remark 6. If h : X → Y is a homeomorphism, then h induces a bijection betweenΩX and ΩY . This justifies our assertion that (X,ΩX) and (Y,ΩY ) should be thoughtof as the same topological space.

Proposition 3.19. Homeomorphism has the following properties.

(1) Both composition and inverse preserve homeomorphisms.

(2) Being homeomorphic is an equivalent relation.

Example 3.20. We have the following.

• h : R→ (0, 1) defined by h(x) = ex/(1 + ex) is a homeomorphism.

• f : [ 0, 1)→ C defined by f(x) = e2πi is continuous but not open.

• In En, the unit ball ∼= the unit cube.

• The stereographic projection h : Sn \ εn+1 → En is a homeomorphism.

• f : E1 → ΩT1 defined by f(x) = x is continuous but is not a homeomorphism.

Figure 3.3. Illustration for the stereographic projection. Stolenfrom Google image.

†School of Mathematics and Statistics, Beijing Institute of Technology, 102488Beijing, P. R. China, ‡Beijing Key Laboratory on MCAACI, Beijing Institute ofTechnology, 102488 Beijing, P. R. China

Email address: kwgl@icloud.com