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{{Stub page|grade=A*|msg=Should be easy to flesh out, find some more references and demote to grade C once acceptable}}
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{{Caution|This page is about topologies only, usually when talk of topologies we don't mean a topology but rather a [[topological space]] which is a topology with its underlying set. See that page for more details}}
==Definition==
A ''topology'' on a [[set]] {{M|X}} is a collection of [[subset|subsets]], {{M|J\subseteq\mathcal{P}(X)}}<ref group="Note">Or {{M|\mathcal{J}\in\mathcal{P}(\mathcal{P}(X))}} if you prefer, here {{M|\mathcal{P}(X)}} denotes the [[power-set]] of {{M|X}}. This means that if {{M|U\in\mathcal{J} }} then {{M|U\subseteq X}}</ref> such that{{rITTMJML}}{{rFAVIDMH}}:
* {{M|X\in\mathcal{J} }} and {{M|\emptyset\in J}}
* If {{M|1=\{U_i\}_{i=1}^n\subseteq\mathcal{J} }} is a [[finite]] collection of elements of {{M|\mathcal{J} }} then {{M|1=\bigcap_{i=1}^nU_i\in\mathcal{J} }} too - {{M|\mathcal{J} }} is [[closed]] under ''[[finite]]'' [[intersection]].
* If {{M|1=\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J} }} is ''any'' collection of elements of {{M|\mathcal{J} }} (finite, [[countable]], [[uncountable]] or otherwise) then {{M|1=\bigcup_{\alpha\in I}U_\alpha\in\mathcal{J} }} - {{M|\mathcal{J} }} is closed under ''[[arbitrary]]'' [[union]].
We call the elements of {{M|\mathcal{J} }} the [[open set|open sets]] of the topology.

A [[topological space]] is simply a [[tuple]] consisting of a set (say {{M|X}}) and a topology (say {{M|\mathcal{J} }}) on that set - {{Top.|X|J}}.
: '''Note: ''' A [[Topology defined in terms of closed sets|topology may be defined in terms of closed sets]] - A [[closed set]] is a subset of {{M|X}} whose [[complement]] is an [[open set]]. A subset of {{M|X}} may be both closed and open, just one, or neither.
==Terminology==
* For {{M|x\in X}} we call {{M|x}} a ''point'' (of the topological space {{Top.|X|J}})<ref name="ITTMJML"/>
* For {{M|U\in\mathcal{J} }} we call {{M|U}} an ''[[open set]]'' (of the topological space {{Top.|X|J}})<ref name="ITTMJML"/>
{{Requires references|just find a glut and spew them here, the definition is the one thing every book I've found agrees on}}
==Examples==
Given a set {{M|X}}, the following topologies can be constructed:
* [[Discrete topology]] - the topology here is {{M|\mathcal{P}(X)}} - the [[power set]] of {{M|X}}. 
* [[Indiscrete topology]] ({{AKA}}: [[Trivial topology]]) - the only open sets are {{M|X}} itself and {{M|\emptyset}}
* [[Finite complement topology]] - the open sets are {{M|\emptyset}} and any set {{M|U\in\mathcal{P}(X)}} such that {{M|\vert X-U\vert\in\mathbb{N} }}
If {{M|(X,d)}} is a [[metric space]], then we have the:
* [[Metric topology]] ({{AKA}}: [[topology induced by a metric]]) - whose open sets are exactly the ones we consider open in a metric sense
** This uses [[open balls]] as a [[topological basis]]
If {{M|(X,\preceq)}} is a [[poset]], then we have the:
* [[Order topology]]
==See also==
* [[Topological separation axioms]]
** Covers things like [[Hausdorff space]], [[Normal topological space]], so forth.
==Notes==
<references group="Note"/>
==References==
<references/>
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{{Definition|Topology|Metric Space}}