nLab separated map

This article is about maps of topological spaces. For morphisms of schemes, see quasiseparated morphism and separated morphism of schemes. For morphisms of toposes, see separated geometric morphisms.

Idea

A family version of the notion of a Hausdorff space.

Definition

A continuous map $f\colon X\to Y$ of topological spaces is separated if the relative diagonal

$$\Delta\colon X\to X\times_Y X$$

is a closed map.

Equivalent formulations:

• The image of the relative diagonal map is a closed subset of $X\times_Y X$.

• Two distinct points of $X$ mapping to the same point of $Y$ have disjoint neighborhoods in $X$.

Properties

Taking $Y$ to be a point, we recover the definition of a Hausdorff space as a space with a closed diagonal map.

Separated maps are closed under base changes.

The point-set formulation of separated maps implies that every fiber of a separated map is a Hausdorff space. The converse is false, since disjoint neighborhoods in a fiber need not come from disjoint neighborhoods in $X$.

Related concepts

• quasiseparated morphism, separated morphism of schemes, separated geometric morphisms

References

• Stacks Project, Tag 0CY0, https://stacks.math.columbia.edu/tag/0CY0.