View source for Evenly covered by a continuous map

{{Stub page|grade=A**|msg=find another source! I have encountered many variations}}
{{Provisional page|grade=A*|msg=Books conflict on whether or not the {{M|V_\alpha}} must be connected.
* By the homeomorphism requirement, the {{M|V_\alpha}} and {{M|U}} have the same number of connected components, that is clear.
* Lee requires the {{M|V_\alpha}} to be connected, that means that {{M|U}} must be connected then. Munkres doesn't. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 21:54, 24 February 2017 (UTC) }}
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==Definition==
Let {{Top.|C|K}} and {{Top.|X|J}} be [[topological spaces]] and let {{M|f:C\rightarrow x}} be a [[continuous map]]. Let {{M|U\in\mathcal{J} }} be given, so {{M|U}} is an [[open set]] of {{Top.|X|J}}, we say that:
* {{M|U}} is ''evenly covered'' by {{M|f}} if{{rITTMJML}}{{rTJRM}}:
** {{MM|\exists \{V_\alpha\}_{\alpha\in I}\subseteq\mathcal{K} }} - there exists an arbitrary collection of open sets - such that:
**# {{MM|f^{-1}(U)\eq\bigcup_{\alpha\in I}V_\alpha}} - the [[union]] of this family is the entire {{link|pre-image|function}} of {{M|U}}<ref group="Note">Note that both sides of the {{M|\eq}} are open:
* Notice that {{M|f^{-1}(U)}} is [[open set|open]] as {{M|f}} is continuous, and that the union of an arbitrary family of open sets is also open, by definition of {{M|\mathcal{K} }} being a [[topology]]</ref>
**# {{M|\forall\alpha,\ \beta\in I[\alpha\neq\beta\implies V_\alpha\cap V_\beta\eq\emptyset]}} - the {{M|V_\alpha}} are [[pairwise disjoint]]
**# {{M|\forall\alpha\in I\big[\big(V_\alpha\cong_{f\vert_{V_\alpha}^\text{Im}:V_\alpha\rightarrow f(V_\alpha)} U\big)\text{ are } }}[[homeomorphism|{{M|\text{homeomorphic} }}]]{{M|\text{ via }f\vert_{V_\alpha}^\text{Im}\big]}} - the [[restriction onto its image]] of {{M|f}} for each {{M|V_\alpha}} is a [[homeomorphism]] onto {{M|U}}
**#* Note that this would mean {{M|\forall\alpha\in I[f(V_\alpha)\eq U]}} 
===Derived constraints===
{{XXX|Notes stage, don't trust yet unless you prove}}
* {{M|U}} and {{M|V_\alpha}} have the same number of [[connected components]] - follows by [[homeomorphism]] part. Must do some theorem about homeomorphisms and components! But I don't want to say "number"
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Algebraic Topology|Topology|Homotopy Theory}}