Definition:Closed 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 a closed set.

Then $P$ is known as an closed invariant.

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

Also see

 * Topological property


 * Continuous invariant
 * Open invariant