Difference between revisions of "Topological space"

Revision as of 20:57, 12 February 2015

A topological space is a set $$X$$ coupled with a topology on $$X$$ denoted $$\mathcal{J}\subset\mathcal{P}(X)$$ , which is a collection of subsets of $$X$$ with the following properties:

1. Both $$\emptyset,X\in\mathcal{J}$$
2. For the collection $$\{U_\alpha\}_{\alpha\in I}\subset\mathcal{J}$$ where $$I$$ is any indexing set, $$\cup_{\alpha\in I}U_\alpha\in\mathcal{J}$$ - that is it is closed under union (infinite, finite, whatever)
3. For the collection $$\{U_i\}^n_{i=1}\subset\mathcal{J}$$ (any finite collection of members of the topology) that $$\cap^n_{i=1}U_i\in\mathcal{J}$$

We write the topological space as $$(X,\mathcal{J})$$ or just $$X$$ if the topology on $$X$$ is obvious.

The elements of $$\mathcal{J}$$ are defined to be "open" sets.