Difference between revisions of "Topological space"
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==Examples== | ==Examples== | ||
Revision as of 01:49, 29 November 2015
TODO: This page is in dire need of an update (was last changed in March 2015, in late Nov 2015 the definition was moved to a subpage, that was it)
Topological space/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.
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]
