Definition:Open Invariant
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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 an open set.
Then $P$ is an open invariant.
Loosely, an open invariant is a property which is preserved in the open image of a mapping.
Also see
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