nLab
perfect space

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.

Definition

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\}.

In a topological space XX, a subset is said to be perfect if it is closed in XX and perfect in its subspace topology.

Example

Properties

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

Reference

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

Last revised on May 9, 2017 at 02:59:12. See the history of this page for a list of all contributions to it.