A description/proof of that 1 point subset of Hausdorff topological space is closed
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of Hausdorff topological space.
- The reader knows a definition of closed set.
- 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 Hausdorff topological space, any 1 point subset 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 Hausdorff topological space, \(T\), any 1 point set, \(\{p\}\), is closed.
2: Proof
Is \(T \setminus \{p\}\) open? For any point, \(p' \in T \setminus \{p\}\), there are some open neighborhoods, \(U_p \subseteq T\) and \(U_{p'} \subseteq T\), of \(p\) and \(p'\) such that \(U_p \cap U_{p'} = \emptyset\). \(U_{p'} \subseteq T \setminus \{p\}\), because for any \(p'' \in U_{p'}\), \(p'' \notin U_p\), \(p'' \notin \{p\}\), so, \(p'' \in T \setminus \{p\}\). So, by the local criterion for openness, \(T \setminus \{p\}\) is open.
