A definition of quotient topology on set with respect to 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 surjection.
Target Context
- The reader will have a definition of quotient topology on set with respect to 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: Definition
For any topological space, \(T\), any set, \(S\), and any surjection, \(f: T \rightarrow S\), the topology on \(S\) defined such that any subset, \(U \subseteq S\), is open if and only \(f^{-1} (U)\) is open
2: Note
The quotient topology is indeed a topology, because \(f^{-1} (\emptyset) = \emptyset\), so, \(\emptyset \subseteq S\) is open; \(f^{-1} (S) = T\), so, \(S \subseteq S\) is open; for any possibly uncountable number of open sets, \(\{U_\alpha\}\), \(f^{-1} (\cup_\alpha U_\alpha) = \cup_\alpha f^{-1} (U_\alpha)\) by the proposition that for any map, the map preimage of any union of sets is the union of the map preimages of the sets, which is open; for any finite number of open sets, \(\{U_i\}\), \(f^{-1} (\cap_i U_i) = \cap_i (f^{-1} (U_i))\) by the proposition that for any map, the map preimage of any intersection of sets is the intersection of the map preimages of the sets, which is open.
