Topological space

Definition

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 $$\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J}$$ 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 $$\{U_i\}^n_{i=1}\subseteq\mathcal{J}$$ (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.

• We call the elements of $\mathcal{J}$ "open sets"

Examples

• Every metric space induces a topology, see the topology induced by a metric space
• Given any set $X$ we can always define the following two topologies on it:
  1. Discrete topology - the topology $\mathcal{J}=\mathcal{P}(X)$ - where $\mathcal{P}(X)$ denotes the power set of $X$
  2. Trivial topology - the topology $\mathcal{J}=\{\emptyset, X\}$

See Also

• Topology
• Topological property theorems