A description/proof of that map of quotient topology is quotient map
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 quotient topology on set with respect to map.
- The reader knows a definition of quotient map.
Target Context
- The reader will have a description and a proof of the proposition that the map of any quotient topology is a quotient map.
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, \(T\), any set, \(S\), any surjection, \(f: T \rightarrow S\), and the quotient topology on \(S\) with respect to \(f\), \(O\), \(f\) is a quotient map.
2: Proof
\(f\) is obviously a continuous surjection by the definition of quotient topology. For any subset, \(U \subseteq S\), if \(f^{-1} (U)\) is open, \(U\) is open by the definition of quotient topology.
