nLab perfect space

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.

Examples

• The empty space is perfect (unless it is excluded by fiat).

• The Cantor space $2^\mathbb{N}$ is perfect, in fact, it is the only totally disconnected, compact, metrizable space that is perfect.

• The real line is perfect.

• Every Polish space (including the previous two examples) is perfect. Furthermore, any closed subset of a Polish space has the perfect-set property.

• Using the axiom of choice, one may prove the existence of subsets of the real line that do not have the perfect-set property. In contrast, one of the axioms of dream mathematics is that every subset of the real line does have this property. This makes the continuum hypothesis a theorem of dream mathematics.

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.

Related entries

• scattered space

• limit point

• Cantor space

• Polish space

• descriptive set theory

Reference

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

category: foundational axiom