A sequence converges in the sence of topological spaces iff the associated statement for filter holds.

The sequential closure of a subset M ⊆ α of a topological space α is the set of all p ∈ α which arise as limit of sequences in M.

A convenience lemma for showing that a set is sequentially closed.

The sequential closure of a set is contained in the closure of that set. The converse is not true.

A set is sequentially closed if it is closed.

The limit of a convergent sequence in a sequentially closed set is in that set.

The limit of a convergent sequence in a closed set is in that set.

A sequential space is a space in which 'sequences are enough to probe the topology'. This can be formalised by demanding that the sequential closure and the closure coincide. The following statements show that other topological properties can be deduced from sequences in sequential spaces.

In a sequential space, a set is closed iff it's sequentially closed.

A function between topological spaces is sequentially continuous if it commutes with limit of convergent sequences.

In a sequential space, continuity and sequential continuity coincide.

Every first-countable space is sequential.