Difference between revisions of "Connected (topology)"

Revision as of 18:21, 19 April 2015 (view source) Alec (Talk | contribs) m ← Older edit		Revision as of 18:53, 19 April 2015 (view source) Alec (Talk | contribs) m (→‎Connected subset) Newer edit →
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	==Connected subset==		==Connected subset==
−	Let {{M|A}} and {{M|B}} be two [[Subspace topology|topological subspaces]] - they are separated if each is disjoint from the [[Closure, interior and boundary|closure]] of the other (closure in {{M|X}}), that is:	+	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>
−	* <math>(B\cap \bar{A})\cup(A\cap\bar{B})=\emptyset</math>	+
	{{Definition|Topology}}		{{Definition|Topology}}

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Revision as of 18:53, 19 April 2015

Definition

A topological space $$(X,\mathcal{J})$$ is connected if there is no separation of $$X$$

Separation

This belongs on this page because a separation is only useful in this definition.

A separation of $$X$$ is a pair of two non-empty open sets $$U,V$$ where $$U\cap V=\emptyset$$ where $$U\cup V=X$$

Equivalent definition

We can also say: 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 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 $\emptyset$^[1]

1. Jump up ↑ Introduction to topology - Mendelson - third edition