Difference between revisions of "Connected (topology)"
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==Connected subset== | ==Connected subset== | ||
A subset {{M|A}} of a [[Topological space]] {{M|(X,\mathcal{J})}} is connected if (when considered with the [[Subspace topology]]) the only two [[Relatively open]] and [[Relatively closed]] (in A) sets are {{M|A}} and {{M|\emptyset}}<ref>Introduction to topology - Mendelson - third edition</ref> | A subset {{M|A}} of a [[Topological space]] {{M|(X,\mathcal{J})}} is connected if (when considered with the [[Subspace topology]]) the only two [[Relatively open]] and [[Relatively closed]] (in A) sets are {{M|A}} and {{M|\emptyset}}<ref>Introduction to topology - Mendelson - third edition</ref> | ||
+ | |||
+ | ==Useful lemma== | ||
+ | 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>U\cap V\subseteq C(A)</math> and | ||
+ | ** Both <math>U\cap A\ne\emptyset</math> and <math>V\cap A\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 | ||
+ | |||
+ | ==Results== | ||
+ | {{Begin Theorem}} | ||
+ | Theorem: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>, <math>U\cap V\subseteq C(A)</math>, <math>U\cap A\ne\emptyset</math> and <math>V\cap A\ne\emptyset</math> | ||
+ | {{Begin Proof}} | ||
+ | {{Todo|Mendelson p115}} | ||
+ | {{End Proof}}{{End Theorem}} | ||
+ | {{Begin Theorem}} | ||
+ | Theorem: The image of a connected set is connected under a continuous map | ||
+ | {{Begin Proof}} | ||
+ | {{Todo|Mendelson p116}} | ||
+ | {{End Proof}} | ||
+ | {{End Theorem}} | ||
+ | |||
+ | |||
+ | ==References== | ||
+ | <references/> | ||
{{Definition|Topology}} | {{Definition|Topology}} |
Revision as of 19:13, 19 April 2015
Contents
Definition
A topological space [math](X,\mathcal{J})[/math] is connected if there is no separation of [math]X[/math]
Separation
This belongs on this page because a separation is only useful in this definition.
A separation of [math]X[/math] is a pair of two non-empty open sets [math]U,V[/math] where [math]U\cap V=\emptyset[/math] where [math]U\cup V=X[/math]
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.
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.
Connected[math]\implies[/math]only sets both open and closed are [math]X,\emptyset[/math]
- Suppose [math]X[/math] is connected and there exists a set [math]A[/math] that is not empty and not all of [math]X[/math] which is both open and closed. Then as :this is closed, [math]X-A[/math] is open. Thus [math]A,X-A[/math] is a separation, contradicting that [math]X[/math] is connected.
Only sets both open and closed are [math]X,\emptyset\implies[/math]connected
TODO:
Connected subset
A subset [ilmath]A[/ilmath] of a Topological space [ilmath](X,\mathcal{J})[/ilmath] is connected if (when considered with the Subspace topology) the only two Relatively open and Relatively closed (in A) sets are [ilmath]A[/ilmath] and [ilmath]\emptyset[/ilmath][1]
Useful lemma
Given a topological subspace [ilmath]Y[/ilmath] of a space [ilmath](X,\mathcal{J})[/ilmath] we say that [ilmath]Y[/ilmath] 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]U\cap V\subseteq C(A)[/math] and
- Both [math]U\cap A\ne\emptyset[/math] and [math]V\cap A\ne\emptyset[/math]
This is definition basically says there has to be a separation of [ilmath]A[/ilmath] that isn't just [ilmath]A[/ilmath] and the [ilmath]\emptyset[/ilmath] for [ilmath]A[/ilmath] to be disconnected, but the sets may overlap outside of A
Results
Theorem:Given a topological subspace [ilmath]Y[/ilmath] of a space [ilmath](X,\mathcal{J})[/ilmath] we say that [ilmath]Y[/ilmath] is disconnected if and only if [math]\exists U,V\in\mathcal{J}[/math] such that: [math]A\subseteq U\cup V[/math], [math]U\cap V\subseteq C(A)[/math], [math]U\cap A\ne\emptyset[/math] and [math]V\cap A\ne\emptyset[/math]
TODO: Mendelson p115
Theorem: The image of a connected set is connected under a continuous map
TODO: Mendelson p116
References
- ↑ Introduction to topology - Mendelson - third edition