Definition:Open Invariant

Definition

Let $P$ be a property whose domain is the set of all topological spaces.

Suppose that whenever $P \left({T \ }\right)$ holds, then so does $P \left({T \ '}\right)$, where:

• $T$ and $T \ '$ are topological spaces
• $\phi \left({T \ }\right) = T \ '$ where $\phi$ is a mapping from $T$ to $T \ '$
• $T \ '$ is an open set.

Then $P$ is known as an open invariant.

Loosely, an open invariant is a property which is preserved in the open image of a mapping.

Also see

• Topological property

• Continuous invariant
• Closed invariant

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Functions