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Connectedness and Disconnectedness in Connectedness and Disconnectedness in topological Spacestopological Spaces

ByBy

Dr. P.K. SharmaDr. P.K. Sharma

P.G.P.G. Department of MathematicsDepartment of MathematicsD.A.V. College , Jalandhar.D.A.V. College , Jalandhar.

(Punjab)(Punjab)

Email : pksharma_davc@yahoo.co.inEmail : pksharma_davc@yahoo.co.in

Connectedness and Disconnectedness in Connectedness and Disconnectedness in topological Spacestopological Spaces

Connectedness is the technical term for the intuitive notion Connectedness is the technical term for the intuitive notion of consisting of a single piece ; in other words, a space X is not of consisting of a single piece ; in other words, a space X is not

connected mean intuitively that it can be split into two non-empty connected mean intuitively that it can be split into two non-empty disjoint parts A and B disjoint parts A and B i.e.i.e. X = A X = A B, A B, A , B , B , A , A B = B = . . If we do not impose any other conditions to A and B, then every If we do not impose any other conditions to A and B, then every space with two or more points can not be connected. This is not space with two or more points can not be connected. This is not consistent, for the familiar spaces like Euclidean spaces, circles, consistent, for the familiar spaces like Euclidean spaces, circles, spheres, intervals (all with usual topologies) be connected so the spheres, intervals (all with usual topologies) be connected so the two parts should not only be mutually disjoint but also that they two parts should not only be mutually disjoint but also that they

should be far away from each other. But the idea of nearness can should be far away from each other. But the idea of nearness can be formalized in terms of closures. This lead us to the following be formalized in terms of closures. This lead us to the following

definition.definition.

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DefinitionDefinition: : A topological space (X , A topological space (X , ) is said to be ) is said to be connected space if it is impossible to find non-empty separated connected space if it is impossible to find non-empty separated subsets A and B such that X = A subsets A and B such that X = A B. B.

i.e.i.e.if A ∄if A ∄ , B , B and A and A B = B = ,,A A B = B = and A and A B = B = X.X.

A space which is not connected is called A space which is not connected is called disconnected space.disconnected space.

Connected SetConnected Set: : Let (X,Let (X, ) be a topological space and A ) be a topological space and A X be X be a subset of X. Then A is called connected set if the subspace a subset of X. Then A is called connected set if the subspace

(A, (A, AA) is connected space.) is connected space.

A set which is not connected is called A set which is not connected is called disconnected set.disconnected set.

RemarkRemark:: Singleton set and empty set are always connected Singleton set and empty set are always connected sets in every topological space.sets in every topological space.

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Theorem:Theorem: A topological space (X, A topological space (X, ) is connected space iff ) is connected space iff

and X are the only sets which are both open as well as closedand X are the only sets which are both open as well as closed..

Theorem:Theorem: A space (X, A space (X, ) is disconnected iff ) is disconnected iff A A ,X and is ,X and is

open as well as closed set.open as well as closed set.

Examples of connected spaces :Examples of connected spaces :

Every indiscrete space is connected space.Every indiscrete space is connected space. Co-finite topological space on infinite set is connected space.Co-finite topological space on infinite set is connected space. Usual topological space is connected space.Usual topological space is connected space. pp-inclusion topological space is connected space.-inclusion topological space is connected space. co-countable topological space on uncountable set X is co-countable topological space on uncountable set X is

connected space.connected space.

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Examples of Disconnected spacesExamples of Disconnected spaces

Every discrete space containing more than one point is Every discrete space containing more than one point is disconnected space.disconnected space.

Lower limit topological space on R, the set of real Lower limit topological space on R, the set of real numbers is disconnected space.numbers is disconnected space.

Sub-space Q of R is disconnected spaceSub-space Q of R is disconnected space

Remark :Remark : (1) Connectedness is not a Hereditary property (1) Connectedness is not a Hereditary property

For exampleFor example : Usual topological space on R is connected : Usual topological space on R is connected space where as subspace Q on R is disconnected spacespace where as subspace Q on R is disconnected space

(2) Disconnectedness is not a Hereditary property(2) Disconnectedness is not a Hereditary property

For exampleFor example :Let X = Y :Let X = Y { x } , where Y is non-empty set { x } , where Y is non-empty set

Then Discrete topological space on X is disconnected Then Discrete topological space on X is disconnected space But its sub-space X- Y , which is indiscrete space space But its sub-space X- Y , which is indiscrete space is connected space . is connected space .

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Connected and disconnected sets in a topological spaceConnected and disconnected sets in a topological space

Intervals and only Intervals are connected sets in usual Intervals and only Intervals are connected sets in usual topological space on R . topological space on R .

In usual topological space on R , the sub-sets In usual topological space on R , the sub-sets ((ii) Q () Q (iiii) Z () Z (iiiiii) N are disconnected sets.) N are disconnected sets.

Properties of Connected set:Properties of Connected set:

Union as well as intersection of two connected sets need Union as well as intersection of two connected sets need not be connected set not be connected set

Subset as well as superset of connected set need not be Subset as well as superset of connected set need not be connected connected

Subset A of a subspace Y is a connected set in Y iff A is Subset A of a subspace Y is a connected set in Y iff A is connected set in the whole space X .connected set in the whole space X .

closure of a connected set is connected set closure of a connected set is connected set If A is connected and A If A is connected and A B B A , then B is connected A , then B is connected

set set

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Continuous image of a connected set is a connected set. Continuous image of a connected set is a connected set. Union of two connected sets having a common point is Union of two connected sets having a common point is

connected.connected. The relation The relation of connectedness in a topological space is an of connectedness in a topological space is an

equivalence relation. i.e. For two points a , b of X , we define equivalence relation. i.e. For two points a , b of X , we define a a b iff a and b are contained in some connected subset of X b iff a and b are contained in some connected subset of X

is an equivalence relation on X .is an equivalence relation on X .

Some Important Theorems in connected space :Some Important Theorems in connected space : A topological space X is connected iff every continuous function A topological space X is connected iff every continuous function

ff from X into the discrete two point space ({0, 1}, D ) is constant. from X into the discrete two point space ({0, 1}, D ) is constant. If be a family of connected sets such that ℱIf be a family of connected sets such that ℱ {{ F : F F : F }, ℱ }, ℱ

, then , then { F : F { F : F } is connected set.ℱ } is connected set.ℱ If E is a connected set in a topological space and E If E is a connected set in a topological space and E A A B, B,

where A and B are separated sets, then either E where A and B are separated sets, then either E A or E A or E B.B.

If every two points of a set E are contained in some connected If every two points of a set E are contained in some connected subset of E, then E is a connected set.subset of E, then E is a connected set.

Arbitrary product of connected spaces is connected space Arbitrary product of connected spaces is connected space

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Component in a topological spaceComponent in a topological space

Definition:Definition: A component E of a topological space is a maximal A component E of a topological space is a maximal connected subset of X. connected subset of X.

Remark : Remark : From the definition, it is clear that a space X is From the definition, it is clear that a space X is connected iff it has only one component namely the whole connected iff it has only one component namely the whole space X.space X.

Examples of componentsExamples of components

Let X = {Let X = {aa, , bb, , cc} and } and = { = { , X, {, X, {cc}, {}, {aa, , bb}} be a topology on }} be a topology on

X.X. Then components of X are {Then components of X are {aa, , bb} and {} and {cc}. }. Singleton sets in discrete topological space, are the only Singleton sets in discrete topological space, are the only

components. components. Singleton sets in Q , the subspace of usual topological space Singleton sets in Q , the subspace of usual topological space

on R are components. on R are components.

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Properties of components in a topological spaceProperties of components in a topological space

Every non-empty connected set is contained in one and only Every non-empty connected set is contained in one and only one componentone component

Every component is closedEvery component is closed Any two distinct components are separated sets Any two distinct components are separated sets Each point of X is contained in one and only one component.Each point of X is contained in one and only one component. Components of topological space form a partition of XComponents of topological space form a partition of X

Some important Theorems in components:Some important Theorems in components: Every non-empty connected subset of X which is both open as Every non-empty connected subset of X which is both open as

well as closed is a component of X.well as closed is a component of X. Continuous image of component need not be component Continuous image of component need not be component Component in a topological space need not an open set.Component in a topological space need not an open set. Component in a totally disconnected space are singleton setsComponent in a totally disconnected space are singleton sets

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Locally connected spacesLocally connected spaces

Definition:Definition:A topological space X is said to be locally connected A topological space X is said to be locally connected at a point at a point xx X iff for every open nhd. U of X iff for every open nhd. U of x x , , a connected a connected open nhd. K of open nhd. K of xx such that such that xx K K U U

In other words X is locally connected at a point In other words X is locally connected at a point xx iff the collection iff the collection of all connected open nhds. of of all connected open nhds. of xx forms a local base at forms a local base at xx..

A space (X, A space (X, ) is said to be ) is said to be locally connected spacelocally connected space iff it is iff it is locally connected at each of its points.locally connected at each of its points.

Examples of Locally connected spaces:Examples of Locally connected spaces: Discrete topological space is locally connected. Discrete topological space is locally connected. Indiscrete topological space is locally connectedIndiscrete topological space is locally connected Usual topological space is locally connectedUsual topological space is locally connected..

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Connectedness and locally connectedness Connectedness and locally connectedness are independent of each otherare independent of each other

X be discrete topological space containing more than one point X be discrete topological space containing more than one point is locally connected which is not connected.is locally connected which is not connected.

Let X = Y Let X = Y {P} , where P is a point (1,0) in plane and Y be the {P} , where P is a point (1,0) in plane and Y be the collection of all points on the line collection of all points on the line yy = (1/n) = (1/n)xx ; ; nn N .N .

Clearly , X is a connected but it is not locally connected .Clearly , X is a connected but it is not locally connected . Properties of locally connected spacesProperties of locally connected spaces Component of a locally connected space is an open set.Component of a locally connected space is an open set. Every open subspace of a locally connected space is locally Every open subspace of a locally connected space is locally

connected.connected. Components of every open subspace of locally connected Components of every open subspace of locally connected

space X are open set.space X are open set. Local connectedness is not a hereditary property.Local connectedness is not a hereditary property. Continuous open image of a locally connected space is locally Continuous open image of a locally connected space is locally

connected. connected.

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Pathwise ConnectivityPathwise Connectivity DefinitionDefinition:(Path):(Path) Let a, b Let a, b X. Then a continuous mapping X. Then a continuous mapping

ff : [0,1] -->X s.t. : [0,1] -->X s.t. ff (0) = (0) = aa and and ff (1) = (1) = bb is called a path from is called a path from aa to to bb.. The point aThe point a is called the initial point and the point is called the initial point and the point bb is called the is called the

terminal point of the path terminal point of the path ff..

RemarkRemark: : If If ff is a path from a to b, then is a path from a to b, then gg:[ 0,1] :[ 0,1] X defined by X defined by gg(x) = (x) = ff(1-x) is a path from b to a . (1-x) is a path from b to a .

Path connected space: Path connected space: A topological space X is called path A topological space X is called path connected iff every two points of X can be join by a path.connected iff every two points of X can be join by a path.

Path connected setPath connected set A subset A of a topological space X is A subset A of a topological space X is called path connected set iff for every two points a and b of A called path connected set iff for every two points a and b of A

a path a path ff : I : IX from X from aa to to bb such that such that ff (I) (I) A . A . Examples of Path connected spaces:Examples of Path connected spaces: Indiscrete space is path connectedIndiscrete space is path connected The n-sphere s(n) = { x The n-sphere s(n) = { x R(n+1): R(n+1): x(2) = 1 } is path connected x(2) = 1 } is path connected

for n >0 .for n >0 .

2R

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Remark (1)Remark (1) Every pathwise connected set is connected. Every pathwise connected set is connected.

(2) Connected set need not be pathwise connected set(2) Connected set need not be pathwise connected set For Example : Let A = { ( 0 , y ) : -1 For Example : Let A = { ( 0 , y ) : -1 y y 1 } and 1 } and

B = {(x,y) : y= sin(1/x) and 0 < x B = {(x,y) : y= sin(1/x) and 0 < x 1 } be subsets of R 1 } be subsets of R R , here A R , here A and B are not separated sets and so Aand B are not separated sets and so AB is connected but there is B is connected but there is no path from a point in A to a point in B.no path from a point in A to a point in B.

Some Important Theorems on Pathwise connected setSome Important Theorems on Pathwise connected set Continuous image of pathwise connected set is pathwise connected Continuous image of pathwise connected set is pathwise connected

set set Product of two path connected spaces is path connectedProduct of two path connected spaces is path connected Union of two intersecting path connected sets in a space is path Union of two intersecting path connected sets in a space is path

connected connected A space X which is connected and locally connected is path A space X which is connected and locally connected is path

connected .connected .

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