A description/proof of that for quotient map, induced map from quotient space of domain by map to codomain is continuous
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 map.
Target Context
- The reader will have a description and a proof of the proposition that for any quotient map, the induced map from the quotient space of the domain by the map to the codomain is continuous.
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 spaces, \(T_1, T_2\), and any quotient map, \(f: T_1 \rightarrow T_2\), the map, \(f': T_1 / f \rightarrow T_2\), is continuous, where \(T_1 / f\) is the quotient space by \(f\), which means that \(f^{-1} (p)\) for each \(p \in T_2\) are identified.
2: Proof
\(f = f' \circ f''\) where \(f'': T_1 \rightarrow T_1 / f\). For any open set, \(U \subseteq T_2\), is \({f'}^{-1} (U)\) open? By the definition of quotient topology, the openness is the openness of \({f''}^{-1} ({f'}^{-1} (U))\), but \({f''}^{-1} ({f'}^{-1} (U)) = (f' \circ f'')^{-1} (U) = f^{-1} (U)\), which is open, as \(f\) is continuous.
