Difference between revisions of "Connected (topology)"

Revision as of 00:42, 22 June 2015

Definition

A topological space is connected if there is no separation of ^[1] A separation of $X$ is:

A pair of non-empty open sets in X, which we'll denote as $U,\ V$ where:
1. $U\cap V=\emptyset$ and
2. $U\cup V=X$

If there is no such separation then the space is connected^[2]

Equivalent definition

This definition is equivalent (true if and only if) the only empty sets that are both open in $X$ are:

$\emptyset$ and
$X$ itself.

I will prove this claim now:

[Expand]
Claim: A topological space $(X,\mathcal{J})$ is connected if and only if the sets $X,\emptyset$ are the only two sets that are both open and closed.

Connected $\implies$ only sets both open and closed are $X,\emptyset$

Suppose

$X$ is connected and there exists a set

$A$ that is not empty and not all of

$X$ which is both open and closed. Then as :this is closed,

$X-A$ is open. Thus

$A,X-A$ is a separation, contradicting that

$X$ is connected.

Only sets both open and closed are $X,\emptyset\implies$ connected

TODO:

Connected subset

A subset $A$ of a Topological space $(X,\mathcal{J})$ is connected if (when considered with the Subspace topology) the only two Relatively open and Relatively closed (in A) sets are $A$ and $\emptyset$ ^[3]

Useful lemma

Given a topological subspace $Y$ of a space $(X,\mathcal{J})$ we say that $Y$ is disconnected if and only if:

$\exists U,V\in\mathcal{J}$ such that:
- $Y\subseteq U\cup V$ and
- $U\cap V\subseteq C(Y)$ and
- Both $U\cap Y\ne\emptyset$ and $V\cap Y\ne\emptyset$

This is basically says there has to be a separation of $Y$ that isn't just $Y$ and the $\emptyset$ for $Y$ to be disconnected, but the sets may overlap outside of {{M|Y}

[Expand]
Proof of lemma:

Results

[Expand]
Theorem:Given a topological subspace $Y$ of a space $(X,\mathcal{J})$ we say that $Y$ is disconnected if and only if $\exists U,V\in\mathcal{J}$ such that: $A\subseteq U\cup V$ , $U\cap V\subseteq C(A)$ , $U\cap A\ne\emptyset$ and $V\cap A\ne\emptyset$

[Expand]
Theorem: The image of a connected set is connected under a continuous map

References

Jump up ↑ Topology - James R. Munkres - 2nd edition
Jump up ↑ Analysis - Part 1: Elements - Krzysztof Maurin
Jump up ↑ Introduction to topology - Mendelson - third edition

[Topology-1] Jump up ↑ Topology - James R. Munkres - 2nd edition

[Analysis-2] Jump up ↑ Analysis - Part 1: Elements - Krzysztof Maurin

[3] Jump up ↑ Introduction to topology - Mendelson - third edition

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
 ==Definition==
-A [[Topological space|topological space]] <math>(X,\mathcal{J})</math> is connected if there is no separation of <math>X</math>
+A [[Topological space|topological space]] <math>(X,\mathcal{J})</math> is connected if there is no separation of <math>X</math><ref name="Topology">Topology - James R. Munkres - 2nd edition</ref> A separation of {{M|X}} is:
+* A pair of non-empty [[Open set|open sets]] in {{M|X}}, which we'll denote as <math>U,\ V</math> where:
-===Separation===
+*# <math>U\cap V=\emptyset</math> and
-This belongs on this page because a separation is only useful in this definition.
+*# <math>U\cup V=X</math>
-A separation of <math>X</math> is a pair of two non-empty [[Open set|open sets]] <math>U,V</math> where <math>U\cap V=\emptyset</math> where <math>U\cup V=X</math>
+If there is no such separation then the space is ''connected''<ref name="Analysis">Analysis - Part 1: Elements - Krzysztof Maurin</ref>
 ==Equivalent definition==
-We can also say: A topological space <math>(X,\mathcal{J})</math> is connected if and only if the sets <math>X,\emptyset</math> are the only two sets that are both open and closed.
+This definition is equivalent (true ''if and only if'') the only empty sets that are both open in {{M|X}} are:
+# {{M|\emptyset}} and
+# {{M|X}} itself.
+I will prove this claim now:
 {{Begin Theorem}}
-Theorem: A topological space <math>(X,\mathcal{J})</math> is connected if and only if the sets <math>X,\emptyset</math> are the only two sets that are both open and closed.
+Claim: A topological space <math>(X,\mathcal{J})</math> is connected if and only if the sets <math>X,\emptyset</math> are the only two sets that are both open and closed.
 {{Begin Proof}}
 '''Connected<math>\implies</math>only sets both open and closed are <math>X,\emptyset</math>'''
@@ Line 27: / Line 29: @@
 Given a [[Subspace topology|topological subspace]] {{M|Y}} of a space {{M|(X,\mathcal{J})}} we say that {{M|Y}} is disconnected '''if and only if''':
 * <math>\exists U,V\in\mathcal{J}</math> such that:
-** <math>A\subseteq U\cup V</math> and
+** <math>Y\subseteq U\cup V</math> and
-** <math>U\cap V\subseteq C(A)</math> and
+** <math>U\cap V\subseteq C(Y)</math> and
-** Both <math>U\cap A\ne\emptyset</math> and <math>V\cap A\ne\emptyset</math>
+** Both <math>U\cap Y\ne\emptyset</math> and <math>V\cap Y\ne\emptyset</math>
-This is definition basically says there has to be a separation of {{M|A}} that isn't just {{M|A}} and the {{M|\emptyset}} for {{M|A}} to be disconnected, but the sets may overlap outside of A
+This is basically says there has to be a separation of {{M|Y}} that isn't just {{M|Y}} and the {{M|\emptyset}} for {{M|Y}} to be disconnected, but the sets may overlap outside of {{M|Y}
+{{Begin Theorem}}
+Proof of lemma:
+{{Begin Proof}}
+{{Todo}}
+{{End Proof}}{{End Theorem}}
 ==Results==

Difference between revisions of "Connected (topology)"

Revision as of 00:42, 22 June 2015

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Definition

Equivalent definition

Connected subset

Useful lemma

Results

References

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