Sequential space: Difference between revisions
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===Symbol-free 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 [[sequence of points]] in the subset having a [[limit]], which lies outside the subset. | 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== | ==Formalisms== | ||
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Here, a sequentially closed subset is a subset that contains the limit of every convergent sequence in it. | Here, a sequentially closed subset is a subset that contains the limit of every convergent sequence in it. | ||
==Relation with | ==Relation with other properties== | ||
===Stronger properties=== | ===Stronger properties=== | ||
* [[ | * [[Radial space]] | ||
* [[First-countable space]] | * [[First-countable space]] | ||
* [[Metrizable space]] | |||
* [[Locally metrizable space]] | |||
==References== | ==References== | ||
Latest revision as of 19:58, 11 May 2008
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