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
[hide]Definition
A topological space (X,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∩V=∅ where U∪V=X
Equivalent definition
We can also say: A topological space (X,J) is connected if and only if the sets X,∅ are the only two sets that are both open and closed.
Theorem: A topological space (X,J) is connected if and only if the sets X,∅ are the only two sets that are both open and closed.
Connected subset
A subset A of a Topological space (X,J) is connected if (when considered with the Subspace topology) the only two Relatively open and Relatively closed (in A) sets are A and ∅[1]
Useful lemma
Given a topological subspace Y of a space (X,J) we say that Y is disconnected if and only if:
- ∃U,V∈J such that:
- A⊆U∪V and
- U∩V⊆C(A) and
- Both U∩A≠∅ and V∩A≠∅
This is definition basically says there has to be a separation of A that isn't just A and the ∅ for A to be disconnected, but the sets may overlap outside of A
Results
Theorem:Given a topological subspace Y of a space (X,J) we say that Y is disconnected if and only if ∃U,V∈J such that: A⊆U∪V, U∩V⊆C(A), U∩A≠∅ and V∩A≠∅
Theorem: The image of a connected set is connected under a continuous map
References
- Jump up ↑ Introduction to topology - Mendelson - third edition
