Open map

Definition

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

Remarks

The projections of a product $X \times Y \to X$ are open maps. A continuous bijection is a homeomorphism if and only if it is an open map. The map $\mathbb{R} \to \mathbb{R}$, $x \mapsto x^2$, is continuous but not open: the image of $(-1, 1)$ is $[0, 1)$.

Explanations

L1 human

An open map keeps open sets open when you push them forward. Continuity is about pulling open sets back, so the two notions are different.