[[Countability axioms]]
# First countability axiom

A topological space $(X, \mathcal{T})$ is **first-countable** iff every $x \in X$ has a countable [[neighbourhood basis]]. #m/def/topology 

Note that since every neighbourhood contains an open neighbourhood,
we can only consider open neighbourhood bases without loss of generality.

## Properties of first-countable spaces

- Every point has a [[Nested neighbourhood basis]].
- [[Sequential continuity]] and [[Continuity]] are equivalent.
- [[Sequential closedness]] and [[Topological space|closedness]] are equivalent.
- [[Conditions for uniqueness of the limit|Uniqueness of the limit]] iff hausdorff.

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