# Difference between revisions of "Topological space"

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# If $\mathcal{O} \subset \tau$ then $\bigcup \mathcal{O} \in \tau$ | # If $\mathcal{O} \subset \tau$ then $\bigcup \mathcal{O} \in \tau$ | ||

# 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. | # 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. | ||

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+ | It should be noted that the [[Open Set|open]] [[Set|set]]s in any [[Metric space|metric space]] satisfy the conditions of a topological space. Therefore every metric space is also a topological space.<ref>Kolmogorov, A. N., and S. V. Fomin. Introductory Real Analysis. Rev. English ed. New York: Dover Publications, 1975. pg. 78-79.</ref> | ||

== Further Reading == | == Further Reading == | ||

+ | * [[Countability Axioms]] | ||

* [[Separation Axioms]] | * [[Separation Axioms]] | ||

== See Also == | == See Also == | ||

+ | * [[Metric space|Metric Space]] | ||

* [[Normal Space]] | * [[Normal Space]] | ||

* [[Regular Space]] | * [[Regular Space]] |

## Revision as of 05:21, 1 December 2018

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]}

- $\emptyset, X \in \tau$
- If $\mathcal{O} \subset \tau$ then $\bigcup \mathcal{O} \in \tau$
- 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.^{[2]}