Topological space
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Definition
A topological space is a set [math]X[/math] coupled with a "topology", [ilmath]\mathcal{J} [/ilmath] on [math]X[/math]. We denote this by the ordered pair [ilmath](X,\mathcal{J})[/ilmath].
 A topology, [ilmath]\mathcal{J} [/ilmath] is a collection of subsets of [ilmath]X[/ilmath], [math]\mathcal{J}\subseteq\mathcal{P}(X)[/math] with the following properties^{[1]}^{[2]}^{[3]}:
 Both [math]\emptyset,X\in\mathcal{J}[/math]
 For the collection [math]\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J}[/math] where [math]I[/math] is any indexing set, [math]\cup_{\alpha\in I}U_\alpha\in\mathcal{J}[/math]  that is it is closed under union (infinite, finite, whatever  "closed under arbitrary union")
 For the collection [math]\{U_i\}^n_{i=1}\subseteq\mathcal{J}[/math] (any finite collection of members of the topology) that [math]\cap^n_{i=1}U_i\in\mathcal{J}[/math]
 We call the elements of [ilmath]\mathcal{J} [/ilmath] "open sets", that is [ilmath]\forall S\in\mathcal{J}[S\text{ is an open set}] [/ilmath], each [ilmath]S[/ilmath] is exactly what we call an 'open set'
As mentioned above we write the topological space as [math](X,\mathcal{J})[/math]; or just [math]X[/math] if the topology on [math]X[/math] is obvious from the context.
Comparing topologies
Given two topological spaces, [ilmath](X_1,\mathcal{J}_1)[/ilmath] and [ilmath](X_2,\mathcal{J}_2)[/ilmath] we may be able to compare them; we say:
Terminology  If  Comment 

[ilmath]\mathcal{J}_1[/ilmath] coarser^{[2]}/smaller/weaker [ilmath]\mathcal{J}_2[/ilmath]  [ilmath]\mathcal{J}_1\subseteq\mathcal{J}_2[/ilmath]  Using the impliessubset relation we see that [ilmath]\mathcal{J}_1\subseteq\mathcal{J}_2\iff\forall S\in\mathcal{J}_1[S\in\mathcal{J}_2][/ilmath] 
[ilmath]\mathcal{J}_1[/ilmath] finer^{[2]}/larger/stronger [ilmath]\mathcal{J}_2[/ilmath]  [ilmath]\mathcal{J}_2\subseteq\mathcal{J}_1[/ilmath]  Again, same idea, [ilmath]\mathcal{J}_2\subseteq\mathcal{J}_2\iff\forall S\in\mathcal{J}_2[S\in\mathcal{J}_1][/ilmath] 
Grade: C
This page requires references, it is on a todo list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
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Need references for larger/smaller/stronger/weaker, Check Introduction To Topology  Mendelson, Addendum: investigate relating this to a poset (easy enough  not very useful / lacking practical applications)
Examples
 Every metric space induces a topology, see the topology induced by a metric space
 Given any set [ilmath]X[/ilmath] we can always define the following two topologies on it:
 Discrete topology  the topology [ilmath]\mathcal{J}=\mathcal{P}(X)[/ilmath]  where [ilmath]\mathcal{P}(X)[/ilmath] denotes the power set of [ilmath]X[/ilmath]
 Trivial topology  the topology [ilmath]\mathcal{J}=\{\emptyset, X\}[/ilmath]
See Also
References
 ↑ Topology  James R. Munkres
 ↑ ^{2.0} ^{2.1} ^{2.2} Introduction to Topological Manifolds  John M. Lee
 ↑ Introduction to Topology  Bert Mendelson
