A description/proof of that open set on open topological subspace is open on base space
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
- The reader knows a definition of subspace topology.
Target Context
- The reader will have a description and a proof of the proposition that any open set on any open topological subspace is open on the base space.
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, \(T_1\), and any open topological subspace, \(T_2 \subseteq T_1\), of \(T_1\), any open set, \(U \subseteq T_2\), on \(T_2\), is open on \(T_1\).
2: Proof
By the definition of subspace topology, \(U = U' \cap T_2\) where \(U' \subseteq T_1\) is open on \(T_1\). As \(T_2\) is open on \(T_1\), \(U\) is open on \(T_1\) as an intersection of finite open sets.
