---
tag: H47ZGRR13V
kind: concept
name: Perfect set property
msc: ["03E15", "54H05"]
sources:
  - url: https://en.wikipedia.org/wiki/Perfect_set_property
same_as:
  wikidata: Q7168098
  wikipedia: Perfect set property
---

## Summary

A subset of a [[Polish space|584GXXY73M]] has the perfect set property if it is either countable or contains a nonempty [[perfect set|628242KN0T]].
A nonempty perfect subset of a Polish space has the [[cardinality of the continuum|0WC28W9DNM]].
So an uncountable set with this property is as large as the whole real line.
Thus the [[continuum hypothesis|0MN2MER9WN]] holds for every class of sets that has the property.
Closed sets, Borel sets and [[analytic sets|3RWWZYPAFV]] have the perfect set property, but the axiom of choice gives sets of reals without it.

## Explanations

### L1 human

The property says that a set of reals cannot be uncountable in a thin or scattered way.
Either the set can be listed as a sequence, or it contains a whole copy of the Cantor set.
For example, the interval $[0, 1]$ is uncountable and is itself perfect: it is closed and none of its points is isolated.
The set of rational numbers is countable, so it has the property for the other reason.
Descriptive set theory asks which definable sets have the property, and the answer depends on the axioms one adopts.