A description/proof of that subspace of 2nd countable topological space is 2nd countable
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 any subspace of any 2nd countable topological space is 2nd countable.
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 2nd countable topological space, \(T\), any subspace, \(T_1 \subseteq T\), is 2nd countable.
2: Proof
\(T\) has a countable basis, \(B\). By the proposition that for any topological space, the intersection of any basis and any subspace is a basis for the subspace, \(T_1\) has the intersection of \(B\) and \(T_1\) as a basis, \(B_1\). As each element of \(B_1\) corresponds to an element of \(B\), \(B_1\) is countable.
