Sequential space

Origin
The notion of sequential space was introduced by S. P. Franklin in 1965.

Symbol-free definition
A topological space is said to be sequential if given any subset of it which is not closed, there is a (possibly transfinite) sequence of points in the subset having a limit, which lies outside the subset.

Formalisms
A sequential space is one where:

sequentially closed subset $$\implies$$ closed subset

Here, a sequentially closed subset is a subset that contains the limit of every convergent sequence in it.

Stronger properties

 * Radial space
 * First-countable space
 * Metrizable space
 * Locally metrizable space