A description/proof of that for metric space, 1 point subset is closed
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of metric space.
- The reader knows a definition of closed subset.
Target Context
- The reader will have a description and a proof of the proposition that for any metric 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 metric space, \(M\), any 1 point subset, \(\{p\} \subseteq M\), is closed.
2: Proof
Think of \(M \setminus \{p\}\) and any point, \(p' \in M \setminus \{p\}\). As \(p \neq p'\), by the definition of distance between 2 points, \(d (p, p') \gt 0\). So, for any \(0 \lt \epsilon \lt d (p, p')\), the open ball around \(p'\), \(B_{p'-\epsilon}\), does not contain \(p\), so, \(B_{p'-\epsilon} \subseteq M \setminus \{p\}\). So, \(M \setminus \{p\}\) is open, so, \(\{p\}\) is closed.
