# 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 == | ||

+ | 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> | ||

+ | |||

+ | == Further Reading == | ||

+ | * [[Topological space|Topological Space]] | ||

+ | |||

+ | == See Also == | ||

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

+ | |||

+ | == References == | ||

+ | <references group="references" /> | ||

+ | |||

+ | [[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

- ↑ Munkres, James R. Topology, 2015. pg. 188.