---
tag: 584GXXY73M
kind: concept
name: Polish space
msc: ["54E50", "54H05", "03E15"]
specializes: [0Z2P41SVDP]
sources:
  - url: https://en.wikipedia.org/wiki/Polish_space
same_as:
  wikidata: Q1207972
  wikipedia: Polish space
---

## Summary

A Polish space is a [[separable|0Z2P41SVDP]] [[topological space|0CQHFPGJ45]] whose topology comes from a [[complete metric|0BY52JW6NK]].
Examples are the real line, $\R^n$, the [[Cantor set|0BXNDZFQME]], the [[Baire space|P2WS16HKD7]] and every separable Banach space.
Polish spaces are the natural setting of [[descriptive set theory|TF8HMVXKYC]] and of much of probability theory.
The name honours the Polish topologists and logicians, such as Sierpiński and Kuratowski, who first studied these spaces in depth.

## Explanations

### L1 human

A Polish space is a space where limits of Cauchy sequences exist and where a countable set of points comes close to every point.
The real line is the model case: the rationals are dense, and Cauchy sequences converge.
The topology matters, not the chosen metric: the open interval $(0, 1)$ is Polish, even though its usual metric is not complete, because it is homeomorphic to the real line.