nLab
Sierpinski space

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Contents

Definition

Definition

The Sierpiński space Σ\Sigma is the topological space

  1. whose underlying set has two elements, say {0,1}\{0,1\},

  2. whose set of open subsets is {,{1},{0,1}}\left\{ \emptyset, \{1\}, \{0,1\} \right\}.

(We could exchange “0” and “1” here, the result would of course be homeomorphic).

Equivalently we may think of the underlying set as the set of of classical truth values {,}\{\bot, \top\}, equipped with the specialization topology, in which {}\{\bot\} is closed and {}\{\top\} is an open but not conversely.

Remark

In constructive mathematics, it is important that {}\{\top\} be open (and {}\{\bot\} closed), rather than the other way around. Indeed, the general definition (since we can't assume that every element is either \top or \bot) is that a subset PP of Σ\Sigma is open as long as it is upward closed: pqp \Rightarrow q and pPp \in P imply that qPq \in P. The ability to place a topology on Top(X,Σ)\Top(X,\Sigma) is fundamental to abstract Stone duality, a constructive approach to general topology.

Properties

As a topological space

This Sierpinski space

According properties are inherited by the Sierpinski topos and the Sierpinski (∞,1)-topos over SierpSierp.

As a classifer for open/closed subspaces

The Sierpinski space SS is a classifier for open subspaces of a topological space XX in that for any open subspace AA of XX there is a unique continuous function χ A:XS\chi_A: X \to S such that A=χ A 1()A = \chi_A^{-1}(\top).

Dually, it classifies closed subsets in that any closed subspace AA is χ A 1()\chi_A^{-1}(\bot). Note that the closed subsets and open subsets of XX are related by a bijection through complementation; one gets a topology on the set of either by identifying the set with Top(X,Σ)\Top(X,\Sigma) for a suitable function space topology. (This part does not work as well in constructive mathematics.)

References

Revised on April 21, 2017 10:28:08 by Urs Schreiber (131.220.184.222)