nLab
sequential topological space

Contents

Idea

A topological space is sequential if (in a certain sense) you can do topology in it using only sequences instead of more general nets.

Sequential spaces are a kind of nice topological space.

Definition

A sequential topological space is a topological space XX such that a subset AA of XX is closed if (hence iff) it contains all the limit points of all sequences of points of AA—or equivalently, such that AA is open if (hence iff) any sequence converging to a point of AA must eventually be in AA.

Equivalently, a topological space is sequential iff it is a quotient space (in TopTop) of a metric space.

Examples

  • Every Frechet–Uryson space is a sequential space.

  • Every topological space satisfying the first countability axiom is Frechet–Uryson, hence a sequential space. In particular, this includes any metrizable space.

  • Every quotient of a sequential space is sequential. In particular, every CW complex is also a sequential space. (Conversely, every sequential space is a quotient of a metrizable space, giving the alternative definition).

Properties

References

  • R. Engelking, General topology

Revised on June 30, 2011 01:36:51 by Toby Bartels (76.85.192.183)