Covering map (topology)

From Maths
Jump to: navigation, search
Stub grade: A**
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Covering space (topology) will be the "main page"
This is a supporting article to the main article: topological covering space


Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](E,\mathcal{ H })[/ilmath] be topological spaces. A map, [ilmath]p:E\rightarrow X[/ilmath] between them is called a covering map[1] if:

  1. [ilmath]\forall U\in\mathcal{J}[p^{-1}(U)\in\mathcal{H}][/ilmath] - in words: that [ilmath]p[/ilmath] is continuous
  2. [ilmath]\forall x\in X\exists e\in E[p(e)\eq x][/ilmath] - in words: that [ilmath]p[/ilmath] is surjective
  3. [ilmath]\forall x\in X\exists U\in\mathcal{O}(x,X)[U\text{ is } [/ilmath][ilmath]\text{evenly covered} [/ilmath][ilmath]\text{ by }p][/ilmath] - in words: for all points there is an open neighbourhood, [ilmath]U[/ilmath], such that [ilmath]p[/ilmath] evenly covers [ilmath]U[/ilmath]

In this case [ilmath]E[/ilmath] is a covering space of [ilmath]X[/ilmath].



  1. Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene