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