View source for Lifting of a continuous map through a covering map

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==Definition==
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| <center><span style="font-size:1.2em;"><m>\xymatrix{ & & E \ar[d]_p \\ Y \ar[rr]^f \ar@{-->}[rru]^\varphi & & X }</m></span></center>
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! If {{M|\varphi}} exists such that all maps are [[continuous]] and the [[diagram commutes]] then {{M|\varphi}} is a ''lifting'' of {{M|f}} ''through'' {{M|p}}
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Let {{Top.|X|J}} be a [[topological space]], let {{Top.|E|H}} be a {{link|covering space|topology}} of {{M|X}}, with {{link|covering map|topology}} {{M|p:E\rightarrow X}}. Then{{rITTMJML}}{{rITTGG}}:
* if we're given a [[continuous map]] {{M|f:Y\rightarrow X}} for an arbitrary topological space {{Top.|Y|K}} such that:
** there exists a continuous map, {{M|\varphi:Y\rightarrow E}}, such that {{M|p\circ\varphi\eq f}} (the diagram on the right [[diagram commutes|commutes]])
*** then {{M|\varphi}} is called a ''lifting'' of {{M|f}} (through {{M|p}})
{{Caveat|I am not sure if we require {{M|Y}} be a [[connected topological space]] or not}}<ref group="Note">Author notes for future use:
* Lee requires that the disjoint elements of an {{link|even covering|topology}} which are [[homeomorphic]] to the [[open neighbourhood]] be themselves [[connected topological space|connected topological (sub)spaces]]
** Lee then requires the {{link|covering space|topology}} be connected and {{link|locally path-connected|topology}}
** He does not require {{M|Y}} to be connected. But for the [[unique lifting property]] he does require {{M|Y}} to be connected
* Gamelin and Greene do not require anything be connected until they reach lifts (here) where {{M|Y}} must be connected.</ref> - however if we do then the [[unique lifting property]] applies.
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==See next==
* [[Unique lifting property]]
* [[Path lifting property]]
* [[Homotopy lifting property]]
* [[Monodromy theorem]]
==Notes==
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==References==
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{{Definition|Topology|Algebraic Topology|Homotopy Theory}}