# Definition:Continuous Invariant

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## 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
$\phi: T \to T'$ is a continuous mapping from $T$ to $T'$
$\phi \left[{T}\right] = T'$, where $\phi \left[{T}\right]$ 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

• Results about continuous invariants can be found here.