Topological space

A topological space is a pair $(X,\tau)$ consisting of a set $X$ and a collection $\tau$ of subsets of $X$, called open sets, satisfying the following axioms:

1. $\emptyset, X \in \tau$
2. If $\mathcal{O} \subset \tau$ then $\bigcup \mathcal{O} \in \tau$
3. If $\{C_1,C_2, \ldots, C_n \}$ is a finite collection of closed sets, then $\bigcap_{n=1}^n C_n $ is a closed set.

It should be noted that the open sets in any metric space satisfy the conditions of a topological space. Therefore every metric space is also a topological space.

Further Reading

• Separation Axioms

See Also

• Metric Space
• Normal Space
• Regular Space
• Kolmogorov Space
• Fréchet Space
• Hausdorff Space
• Topology

External Links

• nLab - Topological Spaces