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 ==
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* [[Countability Axioms]]
 
* [[Separation Axioms]]
 
* [[Separation Axioms]]
  
 
== See Also ==
 
== See Also ==
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* [[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]

  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.[2]

Further Reading

See Also

External Links

References

  1. Steen, Lynn Arthur, and J. Arthur Seebach. Counterexamples in Topology. New York: Dover Publications, 1995. pg. 3.
  2. Kolmogorov, A. N., and S. V. Fomin. Introductory Real Analysis. Rev. English ed. New York: Dover Publications, 1975. pg. 78-79.