Contents

# Contents

## Idea

Georg Cantor used to define the continuum as a perfect space that is connected as well. Hence, the property of ‘being perfect’, as the name indicates as well, can be viewed as forming part of the concept of a ‘prototypical’ topological space.

## Definitions

A topological space $X$ is perfect if it has no isolated points, i.e., if every point $x$ belongs to the topological closure of its complement $X \setminus \{x\}$. Sometimes one requires a perfect space to be inhabited (nonempty), although it is better to allow the empty space.

In a topological space $X$, a subset is said to be perfect if it is closed in $X$ and perfect in its subspace topology. In other words, a set $A$ is perfect if and only if it equals its set $A'$ of accumulation points. Because of the closure requirement, being a perfect set/subset/subspace depends on the ambient space and is stronger than being a perfect space. (But $X$ is a perfect subset of itself iff $X$ is a perfect space.)

In a topological space $X$, a subset $A$ has the perfect-set property if it is either countable (possibly finite or even empty) or has an inhabited perfect subset. Of course, any perfect set has the perfect-set property.

If for each set $B$, ($B$ has a point apart from each point in $A$ if $B$ is inhabited and $B$ is perfect) and ($B$ is empty if $B$ is contained in $A$ and $B$ is perfect) and ($B$ has an isolated point if $B$ is contained in $A$ and inhabited), then $A$ is countable.

## Properties

• Every topological space $X$ is the disjoint union of a scattered subset and a perfect subset.

• Every perfect subset of a Polish space $X$ (including $X$ itself) has the cardinality of the continuum? ($\beth_1$). (This is why the continuum hypothesis follows from the non-classical axiom that every subset of the real line has the perfect-set property.)

• Every closed subset of a Polish space is a unique disjoint union of a countable set and a (possibly empty) perfect set.

## Reference

• S. Willard, General Topology , Addison-Wesley Reading 1970. (Dover reprint 2004, pp.216ff)

Last revised on June 28, 2020 at 05:16:11. See the history of this page for a list of all contributions to it.