Difference between revisions of "Connected (topology)"
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| − | + | 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> | |
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{{Definition|Topology}} | {{Definition|Topology}} | ||
Revision as of 18:53, 19 April 2015
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]- ↑ Introduction to topology - Mendelson - third edition
