Difference between revisions of "Topological space"
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Revision as of 20:57, 12 February 2015
A topological space is a set [math]X[/math] coupled with a topology on [math]X[/math] denoted [math]\mathcal{J}\subset\mathcal{P}(X)[/math], which is a collection of subsets of [math]X[/math] with the following properties:
- Both [math]\emptyset,X\in\mathcal{J}[/math]
- For the collection [math]\{U_\alpha\}_{\alpha\in I}\subset\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)
- For the collection [math]\{U_i\}^n_{i=1}\subset\mathcal{J}[/math] (any finite collection of members of the topology) that [math]\cap^n_{i=1}U_i\in\mathcal{J}[/math]
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.
The elements of [math]\mathcal{J}[/math] are defined to be "open" sets.
