Difference between revisions of "First Countability Axiom"

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The <strong>first countability axiom</strong> is one of two [[Countability Axioms|countability axioms]] related to the classification of [[Topological space|topological space]]s.
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== Definition ==
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A space $X$ is said to have a [[Countable Basis|countable basis]] at $X$ if there is a [[Countable|countable]] collection $\mathscr{B}$ of [[Neighborhood|neighborhood]]s of $X$ such that each neighborhood of $X$ contains at least one of the elements of $\mathscr{B}$. A space that has a countable basis at each of its points is said to satisfy the first countability axiom, or to be ''first-countable''.<ref>Munkres, James R. Topology, 2015. pg. 188.</ref>
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== Further Reading ==
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* [[Topological space|Topological Space]]
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== See Also ==
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* [[Separation Axioms]]
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== References ==
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<references group="references" />
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[[Category:General Topology]]

Latest revision as of 05:20, 1 December 2018

The first countability axiom is one of two countability axioms related to the classification of topological spaces.

Definition

A space $X$ is said to have a countable basis at $X$ if there is a countable collection $\mathscr{B}$ of neighborhoods of $X$ such that each neighborhood of $X$ contains at least one of the elements of $\mathscr{B}$. A space that has a countable basis at each of its points is said to satisfy the first countability axiom, or to be first-countable.[1]

Further Reading

See Also

References

  1. Munkres, James R. Topology, 2015. pg. 188.