View source for Closed map

{{Stub page|grade=B|msg=Created quickly, just to document the concept}}
: A [[open map]] is a thing to and is defined similarly. 
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==Definition==
Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[map]] (not necessarily [[continuous]] - just a map between {{M|X}} and {{M|Y}} considered as sets), then {{nowrap|we call {{M|f}} a ''closed map'' if{{rITTMJML}}}}:
* {{M|1=\forall U\in C(X,\mathcal{J})[f(U)\in C(Y,\mathcal{K})]}} - that is, that the {{link|image|map|s}} (under {{M|f}}) of all {{plural|closed set|s}} of {{Top.|X|J}} are [[closed set|closed]] in {{Top.|Y|K}}
** {{M|C(X,\mathcal{J})}} denotes the set of all {{plural|closed set|s}} of the [[topological space]] {{Top.|X|J}}
===Reformulation===
A set is [[closed set|closed]] if its [[complement]] is [[open set|open]], so we could state:
* A [[mapping]] {{M|f:X\rightarrow Y}} is a ''closed map'' if:
** {{M|1=\forall E\in\mathcal{P}(X)[(X-E)\in\mathcal{J}\implies (Y-f(E))\in\mathcal{K}]}} - this is '''Claim 1'''
See proof of claims below.
: '''Note: ''' we really have an {{iff}} relationship here. See [[definitions and iff]] for information
==Proof of claims==
===Claim 1: Reformulation===
{{Requires proof|easy=true|grade=D|msg=Easy proof, skipped. Proof would be that a map is closed WRT the definition {{iff}} it satisfies the reformulation}}
==References==
<references/>
{{Definition|Topology}}