# Definition:Topological Property

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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 which are homeomorphic.

Then $P$ is known as a **topological property**.

Loosely, a topological property is one which is preserved under homeomorphism.

## Also known as

- A
**topological property**is also known as a**topological invariant**.

## Also see

- Results about
**topological properties**can be found**here**.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.6$: Homeomorphisms - 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 - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$)