nLab
sequential topological space

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

Equivalently, a topological space is sequential iff it is a quotient space (in $\mathrm{Top}$) 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

• The category of sequential spaces is a coreflective subcategory of the category of all topological spaces.

• The category of sequential spaces is a reflective subcategory of the category of subsequential spaces, much as $\mathrm{Top}$ itself is a reflective subcategory of the category of all pseudotopological spaces.

• The category of sequential spaces is cartesian closed. See also convenient category of topological spaces.

References

• R. Engelking, General topology