Open Map

A function $f:X\to Y$ between Topological Spaces is an open map if the Direct Image $f_{*}(U)$ of an open set $U\in\mathcal{O}({X})$ is open in $Y$.

Examples

• The projections $\pi_{i}:\prod_{i:I}X_{i}\to X_{i}$ are open maps with respect to the Product Topology.

• More generally, projections out of Fibre Bundles are open maps.

• Every map $X\to Y$ is open when we equip $Y$ with the Discrete Topology, even though it may not be Continuous.

Properties

• Open maps are closed under composition, and every Homeomorphism is an open map.

• A bijective continuous map is a Homeomorphism if and only if it is open.