View source for Notes:Covering spaces

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==Gamelin & Greene==
* '''Evenly covered'''
** Let {{M|p:E\rightarrow X}} be [[continuous map]], we say {{M|U}} open in {{M|X}} is ''evenly covered'' by {{M|p}} if:
*** {{M|p^{-1}(U)}} equals a union of disjoint open sets of {{M|E}} such that each one of these disjoint open sets is [[homeomorphic]] onto {{M|U}} if you restrict {{M|p}} to it
* '''Covering map'''
** {{M|p:E\rightarrow X}} is a covering map if:
**# {{M|p}} is [[surjective]]
**# {{M|\forall x\in X\exists U\in\mathcal{O}(x,X)[U\text{ evenly covered by }p]}}
** We say "{{M|E}} is a covering space of {{M|X}}"
===Lift of a continuous map===
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| <center><span style="font-size:1.2em;"><m>\xymatrix{ & & E \ar[d]^p \\ Y \ar@{.>}[rru]^g \ar[rr]^f & & X }</m></span></center>
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Let {{M|p:E\rightarrow X}} be a covering map, let {{Top.|Y|K}} be a [[topological space]] and let {{M|f:Y\rightarrow X}} be a [[continuous map]]
* If there is a map: {{M|g:Y\rightarrow E}} such that {{M|p\circ g\eq f}} then {{M|g}} is called a ''lift'' of {{M|f}}
{{ie}} if the diagram on the right commutes
===Claims===
# '''Uniqueness of lifts'''
#* Let {{M|p:E\rightarrow X}} be a covering map, let {{M|Y}} be a ''{{link|connected|topology}}'' [[topological space]] and let {{M|f:Y\rightarrow X}} be a [[continuous map]].
#** Let {{M|g,h:Y\rightarrow E}} be two ''lifts'' of {{M|f}}
#*** if {{M|\exists y\in Y[g(y)\eq h(y)]}} then {{M|g\eq y}}
#* Take some time to think about this, it's basically saying if they're lifts over the same part then they agree, the diagram above commuting has large implications
# '''Path lifting theorem'''
#* Let {{M|p:E\rightarrow X}} be a covering map, let {{M|\gamma\in}}[[C(I,X)|{{M|C([0,1],X)}}]] (a {{link|path|topology}}), then as {{M|p}} is a covering map {{M|\exists e_0\in E[\gamma(0)\eq p(e_0)]}}, we claim:
#** There is a unique path (or "lift" of {{M|\gamma}}): {{M|\alpha:[0,1]\rightarrow E}} such that {{M|\alpha(0)\eq e_0}} and such that {{M|p\circ\alpha\eq\gamma}}
#* {{Caveat|Note that if {{M|\gamma}} was a loop that {{M|\alpha}} need not be a loop!}}
# '''Path-homotopy lifting theorem''' (named by Alec)
#* Let {{M|p:E\rightarrow X}} be a covering map and let {{M|F:[0,1]\times[0,1]\rightarrow X}} be a [[homotopy]] (a continuous map in this case), then by the covering map: {{M|\exists e_0\in E[p(e_0)\eq F(0,0)]}}
#** Then we claim there exists a unique lift of {{M|F}}, say {{M|G:[0,1]\times[0,1]\rightarrow E}} such that {{M|G(0,0)\eq e_0}}
==Proof==
===Uniqueness of lifts===
* We want to show {{M|\big(\exists y_0\in Y[g(y_0)\eq h(y_0)]\big)\implies g\eq h}}, to show {{M|g\eq h}} we must show: {{M|\forall y\in Y[g(y)\eq h(y)]}}