Topological space

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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]:
  1. Both [math]\emptyset,X\in\mathcal{J}[/math]
  2. 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")
  3. 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 implies-subset 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 to-do list for being expanded with them.
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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)


See Also


  1. Topology - James R. Munkres
  2. 2.0 2.1 2.2 Introduction to Topological Manifolds - John M. Lee
  3. Introduction to Topology - Bert Mendelson