A definition of covering map
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of connected topological space.
- The reader knows a definition of locally path-connected topological space.
- The reader knows a definition of surjection.
- The reader knows a definition of continuous map.
- The reader knows a definition of neighborhood of point.
- The reader knows a definition of homeomorphism.
Target Context
- The reader will have a definition of covering 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 connected and locally path-connected topological spaces, \(T_1, T_2\), any surjective and continuous map, \(\pi: T_1 \to T_2\), such that for any point, \(p \in T_2\), there is a neighborhood, \(N_p \subseteq T_2\), that is evenly covered by \(\pi\), which means that for each connected-component of \(\pi^{-1} (N_p)\), \(\pi^{-1} (N_p)_\alpha\), where \(\alpha \in A_p\) where \(A_p\) is a possibly uncountable indices set, \(\pi\vert_{\pi^{-1} (N_p)_\alpha}: \pi^{-1} (N_p)_\alpha \to N_p\) is a homeomorphism; \(T_2\) is called base of covering, \(T_1\) is called covering space of \(T_2\), each \(\pi^{-1} (N_p)_\alpha\) is called sheet of covering over \(N_p\)
