Sequential space

A topological space $X$ such that if $A\subset X$ and $A\neq[A]$ (that is, the set $A$ is not closed), then there is a sequence $x_k$, $k=1,2,\dots,$ of points of $A$ that converges to a point of $[A]\setminus A$. If $x\in[A]\subset X$ always implies that there is a sequence $x_k$ of points from $A$ that converges to $x$, then $X$ is called a Fréchet–Urysohn space.

Comments

Sequential spaces form a coreflective subcategory (see Reflective subcategory) of the category of all topological spaces; the coreflection is obtained by retopologizing an arbitrary space with the topology in which a subset is closed if and only if it is closed under limits of sequences (in the original topology). Spaces which satisfy the first axiom of countability are always sequential (indeed, they are Fréchet–Urysohn spaces), and the sequential spaces form the smallest coreflective subcategory containing all first-countable spaces. For this reason, many topological results which are traditionally proved for first-countable spaces can readily be extended to sequential spaces.

References

[a1]	R. Engelking, "General topology" , Heldermann (1989)