Topological property

Summary

A topological property is a property of topological spaces such that, if a space has it, each homeomorphic space also has it. Examples are compactness, connectedness, separability, metrizability and the Hausdorff property.

Remarks

Topological properties are used to show that two spaces are not homeomorphic. For example, $[0, 1]$ is compact and $(0, 1)$ is not, so they are not homeomorphic. Properties that depend on a metric are often not topological. For example, $(0, 1)$ and $\mathbb{R}$ are homeomorphic, but only $\mathbb{R}$ is complete and only $(0, 1)$ is bounded.

Explanations

L1 human

A topological property is a property that a space keeps when you deform it by a homeomorphism. Size and distance are not topological properties, but the number of pieces of a space is one.