A description/proof of that connected component is open on locally connected topological space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of locally connected topological space.
- The reader admits the local criterion for openness.
Target Context
- The reader will have a description and a proof of the proposition that for any locally connected topological space, each connected component is open.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any locally connected topological space, \(T\), each connected component, \(S\), is open.
2: Proof
At any point, \(p \in S\), take any neighborhood of \(p\), \(U_p\), on \(T\). There is a connected neighborhood of \(p\), \({U_p}'\), contained in \(U_p\). As \({U_p}'\) is a connected subspace that contains \(p\), \({U_p}' \subseteq S\). By the local criterion for openness, \(S\) is open.
