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: Structured Description
Here is the rules of Structured Description.
Entities:
\( T_1\): \(\in \{\text{ the connected and locally path-connected topological spaces }\}\)
\( T_2\): \(\in \{\text{ the connected and locally path-connected topological spaces }\}\)
\(*\pi\): \(: T_1 \to T_2\), \(\in \{\text{ the surjections }\} \cap \{\text{ the continuous maps }\}\)
//
Conditions:
\(\forall t \in T_2 (\exists U_t \subseteq T_2 \in \{\text{ the open neighborhoods of } t\} \text{ such that } U_t \text{ is evenly-covered by } \pi)\), where "evenly-covered by \(\pi\)" means that for each connected component of \(\pi^{-1} (U_t)\), \(\pi^{-1} (U_t)_j\), where \(j \in J\) where \(J\) is a possibly uncountable index set, \(\pi \vert_{\pi^{-1} (U_t)_j}: \pi^{-1} (U_t)_j \to U_t\) is a homeomorphism
//
\(T_2\) is called "base of covering".
\(T_1\) is called "covering space of \(T_2\)".
Each \(\pi^{-1} (U_t)_j\) is called "sheet of covering over \(U_t\)".
