Sequential continuity

A function between topological spaces is sequentially continuous at a point iff any convergent sequence in with limit maps to a convergent sequence in with limit , i.e.

All continuous maps are sequentially continuous. In case is first-countable, then is sequentially continuous at a point iff. it is continuous at that point. #m/thm/topology

Proof for first-countable equivalence

Let and be topological space, be a map, and be a point. Let , and be first-countable.

First, assume is continuous at . Let be a sequence in with , and . Then , and thus there exists such that for all , whence for all . Therefore , without invoking the First countability axiom.

For the converse, assume is sequentially continuous at . Let a countable nested open neighbourhood basis of . Assume is not continuous at , i.e. there exists such that for all . We can then construct a sequence such that for all , where clearly , but for all . whence , contradicting our requirement that be sequentially continuous. Therefore, is continuous at .

Another topological property that can be shown using sequences for metric spaces is Sequential closedness.

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