nLab
perfect space

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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 XX is perfect if it has no isolated points, i.e., if every point xx belongs to the topological closure of its complement X{x}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 XX, a subset is said to be perfect if it is closed in XX and perfect in its subspace topology. In other words, a set AA is perfect if and only if it equals its set AA' 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 XX is a perfect subset of itself iff XX is a perfect space.)

In a topological space XX, a subset AA 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 BB, (BB has a point apart from each point in AA if BB is inhabited and BB is perfect) and (BB is empty if BB is contained in AA and BB is perfect) and (BB has an isolated point if BB is contained in AA and inhabited), then AA is countable.

Examples

Properties

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

  • Every perfect subset of a Polish space XX (including XX itself) has the cardinality of the continuum? ( 1\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.