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 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
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