# 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
$\phi: T \to T'$ is a mapping from $T$ to $T'$
$\phi \left[{T}\right] = T'$, where $\phi \left[{T}\right]$ denotes the image of $\phi$
$T'$ is a closed set.

Then $P$ is a closed invariant.

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