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

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Functions