Sequential space

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

History

Origin

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

Definition

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

Subspace property implication formalism

This property of topological spaces can be encoded by the fact that one subspace property implies another

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.

Relation with other properties

Stronger properties

• Radial space
• First-countable space

References

• Spaces in which sequences suffice by S. P. Franklin, Fund. Math. 57 (1965), 107-115
• Spaces in which sequences suffice II by S. P. Franklin, Fund. Math. 61 (1967), 51-56

External links

• Wikipedia page