Vipul: Created page with '==Statement== Suppose $E$ and $B$ are topological spaces and $p:E \to B$ is a fact about::covering map[[uses property satisfaction of::c...'

2010-12-25T02:03:03Z

Created page with '==Statement== Suppose <math>E</math> and <math>B</math> are topological spaces and <math>p:E \to B</math> is a fact about::covering map[[uses property satisfaction of::c...'

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==Statement==

Suppose <math>E</math> and <math>B</math> are [[topological space]]s and <math>p:E \to B</math> is a [[fact about::covering map]][[uses property satisfaction of::covering map| ]]. Then, <math>p</math> is a [[fact about::local homeomorphism]][[proves property satisfaction of::local homeomorphism| ]].

==Proof==

'''Given''': A covering map <math>p:E \to B</math>. A point <math>e \in E</math>.

'''To prove''': There exists an open subset <math>V</math> of <math>E</math> containing <math>e</math> such that the restriction of <math>p</math> to <math>V</math> is a homeomorphism.

'''Proof''':

{| class="sortable" border="1"
! Step no. !! Assertion !! Definitions used !! Previous steps used !! Explanation
|-
| 1 || Let <math>b = p(e)</math> || -- || -- ||
|-
| 2 || There exists an open subset <math>U</math> of <math>B</math> such that <math>b \in U</math>, a discrete space <math>F</math>, and a homeomorphism <math>\varphi:U \times F \to p^{-1}(U)</math> such that <math>p \circ \varphi</math> is projection on the first coordinate. || [[covering map]] || (1) ||
|-
| 3 || Let <math>f</math> be the second coordinate of <math>\varphi^{-1}(e)</math>. Let <math>V</math> be <math>\varphi(U \times \{ f \})</math>. Then <math>V</math> contains <math>e</math> || -- || (2) || By definition, <math>\varphi^{-1}(e) \in U \times \{ f \}</math>. Since <math>\varphi</math> is bijective, we get that <math>e \in \varphi(U \times \{ f \})</math>.
|-
| 4 || <math>V</math> is an open subset of <math>p^{-1}(U)</math> || -- || (2), (3) || Since <math>\varphi</math> is a homeomorphism, it suffices to note that <math>U \times \{ f \}</math> is an open subset of <math>U \times F</math>. This in turn follows from the fact that <math>F</math> being a discrete space, <math>\{ f \}</math> is an open subset of <math>F</math>.
|-
| 5 || <math>V</math> is an open subset of <math>E</math> || -- || (2), (4) || Since <math>p</math> is continuous and <math>U</math> is open in <math>B</math>, <math>p^{-1}(U)</math> is open in <math>E</math>. Step (4) says that <math>V</math> is open in <math>p^{-1}(U)</math>. Combining, we get that <math>V</math> is open in <math>E</math>.
|-
| 6 || The restriction of <math>p</math> to <math>V</math> gives a homeomorphism from <math>V</math> to <math>U</math>. || -- || (2)-(5) || We have that <math>p = (p \circ \varphi) \circ \varphi^{-1}</math>. Restricted to <math>V</math>, the map is the composite of a homeomorphism from <math>V</math> to <math>U \times \{ f \}</math> and a projection (homeomorphism) from <math>U \times \{ f \}</math> to <math>U</matH>. Hence, the map overall is a homeomorphism.
|}

Covering map implies local homeomorphism - Revision history

Vipul: Created page with '==Statement== Suppose E and B are topological spaces and p:E \to B is a fact about::covering map[[uses property satisfaction of::c...'

Vipul: Created page with '==Statement== Suppose $E$ and $B$ are topological spaces and $p:E \to B$ is a fact about::covering map[[uses property satisfaction of::c...'