Definition:Open Mapping

Definition

Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be topological spaces.

Let $f: S_1 \to S_2$ be a mapping.

Then $f$ is said to be an open mapping if and only if:

$\forall U \in \tau_1: f \sqbrk U \in \tau_2$

where $f \sqbrk U$ denotes the image of $U$ under $f$.

Also see

• Definition:Closed Mapping

• Definition:Continuous Mapping (Topology), a different concept not to be confused with this

• Results about open mappings can be found here.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Functions
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): open mapping
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): open mapping