Editing Covering space (section)

== Definition ==

Let <math>X</math> be a topological space. A '''covering''' of <math>X</math> is a continuous map
: <math>\pi : \tilde X \rightarrow X</math>
such that for every <math>x \in X</math> there exists an [[Neighbourhood (mathematics)|open neighborhood]] <math>U_x</math> of <math>x</math> and a [[discrete space]] <math>D_x</math> such that <math>\pi^{-1}(U_x)= \displaystyle \bigsqcup_{d \in D_x} V_d </math> and <math>\pi|_{V_d}:V_d \rightarrow U_x </math> is a [[homeomorphism]] for every <math>d \in D_x </math>.
The open sets <math>V_{d}</math> are called '''sheets''', which are uniquely determined up to homeomorphism if <math>U_x</math> is [[Connected space|connected]].{{r|Hatcher|p=56}} For each <math>x \in X</math> the discrete set <math>\pi^{-1}(x)</math> is called the '''fiber''' of <math>x</math>. If <math>X</math> is connected (and <math>\tilde X</math> is non-empty), it can be shown that <math>\pi</math> is [[surjective]], and the [[cardinality]] of <math>D_x</math> is the same for all <math>x \in X</math>; this value is called the '''degree''' of the covering. If <math>\tilde X</math> is [[Path connected|path-connected]], then the covering <math> \pi : \tilde X \rightarrow X</math> is called a '''path-connected covering'''. This definition is equivalent to the statement that <math>\pi</math> is a locally trivial [[Fiber bundle]].

Some authors also require that <math>\pi</math> be surjective in the case that <math>X</math> is not connected.<ref>Rowland, Todd. "Covering Map." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CoveringMap.html</ref>