Closed map

Definition

Let $X$ and $Y$ be topological spaces. A function $f \colon X \to Y$ is a closed map when for each closed set $F$ of $X$, the image $f(F)$ is closed in $Y$.

Remarks

A continuous map from a compact space to a Hausdorff space is closed. This gives continuous bijections from compact to Hausdorff spaces are homeomorphisms. The projection $\mathbb{R}^2 \to \mathbb{R}$, $(x, y) \mapsto x$, is open but not closed: it maps the closed set $\{(x, y) : xy = 1\}$ to $\mathbb{R} \setminus \{0\}$. A map can be closed but not open. A constant map $\mathbb{R} \to \mathbb{R}$ is closed, but the image of $\mathbb{R}$ is a point, which is not open.

Explanations

L1 human

A closed map keeps closed sets closed when you push them forward. Open maps and closed maps are different: a map can have one property without the other.