Topological space

Definition

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:

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)
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 write the topological space as [math](X,\mathcal{J})[/math] or just [math]X[/math] if the topology on [math]X[/math] is obvious.

• We call the elements of [ilmath]\mathcal{J} [/ilmath] "open sets"

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:
  1. 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]
  2. Trivial topology - the topology [ilmath]\mathcal{J}=\{\emptyset, X\}[/ilmath]

See Also

• Topology
• Topological property theorems