Definition:Continuous 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 continuous mapping from $T$ to $T'$

$\phi \sqbrk T = T'$, where $\phi \sqbrk T$ denotes the image of $\phi$.

Then $P$ is a continuous invariant.

Loosely, a continuous invariant is a property which is preserved under a continuous mapping.

Also see

• Definition:Topological Property

• Definition:Open Invariant
• Definition:Closed Invariant

• Results about continuous invariants can be found here.

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