Topology
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 [ilmath]X[/ilmath] is a collection of subsets, [ilmath]J\subseteq\mathcal{P}(X)[/ilmath]^{[Note 1]} such that^{[1]}^{[2]}:
 [ilmath]X\in\mathcal{J} [/ilmath] and [ilmath]\emptyset\in J[/ilmath]
 If [ilmath]\{U_i\}_{i=1}^n\subseteq\mathcal{J}[/ilmath] is a finite collection of elements of [ilmath]\mathcal{J} [/ilmath] then [ilmath]\bigcap_{i=1}^nU_i\in\mathcal{J}[/ilmath] too  [ilmath]\mathcal{J} [/ilmath] is closed under finite intersection.
 If [ilmath]\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J}[/ilmath] is any collection of elements of [ilmath]\mathcal{J} [/ilmath] (finite, countable, uncountable or otherwise) then [ilmath]\bigcup_{\alpha\in I}U_\alpha\in\mathcal{J}[/ilmath]  [ilmath]\mathcal{J} [/ilmath] is closed under arbitrary union.
We call the elements of [ilmath]\mathcal{J} [/ilmath] the open sets of the topology.
A topological space is simply a tuple consisting of a set (say [ilmath]X[/ilmath]) and a topology (say [ilmath]\mathcal{J} [/ilmath]) on that set  [ilmath](X,\mathcal{ J })[/ilmath].
 Note: A topology may be defined in terms of closed sets  A closed set is a subset of [ilmath]X[/ilmath] whose complement is an open set. A subset of [ilmath]X[/ilmath] may be both closed and open, just one, or neither.
Terminology
 For [ilmath]x\in X[/ilmath] we call [ilmath]x[/ilmath] a point (of the topological space [ilmath](X,\mathcal{ J })[/ilmath])^{[1]}
 For [ilmath]U\in\mathcal{J} [/ilmath] we call [ilmath]U[/ilmath] an open set (of the topological space [ilmath](X,\mathcal{ J })[/ilmath])^{[1]}
The message provided is:
Examples
Given a set [ilmath]X[/ilmath], the following topologies can be constructed:
 Discrete topology  the topology here is [ilmath]\mathcal{P}(X)[/ilmath]  the power set of [ilmath]X[/ilmath].
 Indiscrete topology (AKA: Trivial topology)  the only open sets are [ilmath]X[/ilmath] itself and [ilmath]\emptyset[/ilmath]
 Finite complement topology  the open sets are [ilmath]\emptyset[/ilmath] and any set [ilmath]U\in\mathcal{P}(X)[/ilmath] such that [ilmath]\vert XU\vert\in\mathbb{N} [/ilmath]
If [ilmath](X,d)[/ilmath] 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 [ilmath](X,\preceq)[/ilmath] is a poset, then we have the:
See also
 Topological separation axioms
 Covers things like Hausdorff space, Normal topological space, so forth.
Notes
 ↑ Or [ilmath]\mathcal{J}\in\mathcal{P}(\mathcal{P}(X))[/ilmath] if you prefer, here [ilmath]\mathcal{P}(X)[/ilmath] denotes the powerset of [ilmath]X[/ilmath]. This means that if [ilmath]U\in\mathcal{J} [/ilmath] then [ilmath]U\subseteq X[/ilmath]
References
 ↑ ^{1.0} ^{1.1} ^{1.2} Introduction to Topological Manifolds  John M. Lee
 ↑ Functional Analysis  Volume 1: A gentle introduction  Dzung Minh Ha
