A description/proof of that connected component is closed
Topics
About: topological space
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that for any topological space, each connected component is closed.
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 topological space, \(T\), each connected component, \(S\), is closed.
2: Proof
Suppose that \(S\) was not closed. By the proposition that for any topological space and any connected subspace, any subspace that contains the connected subspace and is contained in the closure of the connected subspace is connected, the closure of \(S\), \(\overline{S}\), is connected, so, the connected component would not be \(S\) but larger than \(S\), a contradiction. So, \(S\) is closed.
