nLab
scattered 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

A scattered space is a topological space that on the scale of ‘connectivity’ is at the extreme opposite of perfect spaces.

Scattered spaces are used to provide topological models of provability logic.

Definition

A subset AA of a topological space is called dense in itself if each point of AA is a limit point of AA.

A topological space XX is called scattered if XX doesn’t contain nonempty dense-in-itself subsets.

Example

  • The Sierpinski space X={a,b}X=\{a,b\} with topology {,{a},X}\{\emptyset ,\{a\}, X\} is scattered: whereas bb is a limit point of XX, aa isn’t, whence XX fails to be dense in itself. {a}\{a\} fails to be dense in itself since, whereas bb is a limit point, aa isn’t. Finally, {b}\{b\} has no limit points at all.

  • Discrete spaces are scattered.

Properties

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

Remark

Given a space XX, using the derivative :2 X2 X\partial : 2^X\to 2^X that maps a subset to the set of its limit points, the property of being dense-in-itself amounts to AAA\subseteq\partial A. Sierpiński (1927) studies the abstract properties of dense-in-itself and corresponding scattered subsets relative to arbitrary monotone maps 2 X2 X2^X\to 2^X.

Reference

  • L. Beklemishev, D. Gabelaia, Topological interpretations of provability logic , arXiv:1210.7317 (2012). (abstract)

  • W. Sierpiński, La notion de derivée comme base d’une théorie des ensembles abstraits , Math. Ann. 97 (1927) pp.321-337. (gdz)

  • S. Willard, General Topology , Addison-Wesley Reading 1970. (Dover reprint 2004, p.219)

Revised on June 16, 2016 06:45:13 by Thomas Holder (176.4.28.225)