Definition:Closed Invariant

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

Suppose that whenever $\map P T$ holds, then so does $\map P {T'}$, where:

$T$ and $T'$ are topological spaces
$\phi: T \to T'$ is a mapping from $T$ to $T'$
$\phi \sqbrk T = T'$, where $\phi \sqbrk T$ 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.

Also see