---
tag: 02SMVGC690
kind: concept
name: Topological property
aliases: [topological-invariant]
msc: [54A05]
sources:
  - url: https://en.wikipedia.org/wiki/Topological_property
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Topological_property
    locator: Non-topological properties
same_as:
  wikidata: Q625948
  wikipedia: Topological property
---

## Summary

A topological property is a property of topological spaces such that, if a space has it, each [[homeomorphic|05GZ11PWRY]] space also has it.
Examples are [[compactness|0QPHW703PV]], [[connectedness|0D939CMME7]], [[separability|0GKJ78948V]], [[metrizability|09KDHB82SG]] and the [[Hausdorff property|0MXB4CKHFW]].

## 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|08EWT7J0SC]] 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.