Definition:Closed Invariant

Definition
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

 * Definition:Topological Property


 * Definition:Continuous Invariant
 * Definition:Open Invariant