nLab
Sierpinski space

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Context

Topology

topology

algebraic topology

Basic concepts

Theorems

Extra stuff, structure, properties

Examples

Contents

Definition

Definition

The Sierpiński space Σ is the topological space which is the set of truth values, classically {⊥,⊤}, equipped with the specialization topology, in which {⊥} is closed and {⊤} is open but not conversely. (The opposite convention is also used.)

Remark

In constructive mathematics, it is important that {⊤} be open (and {⊥} closed), rather than the other way around. Indeed, the general definition (since we can't assume that every element is either ⊤ or ⊥) is that a subset P of Σ is open as long as it is upward closed: p⇒q and p∈P imply that q∈P. The ability to place a topology on Top(X,Σ) 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 Sierp.

As a classifer for closed subspaces

The Sierpinski space is a classifier for closed subspaces of a topological space X in that for any closed subspace A of X there is a unique continuous function χ A:X→S such that A=χ A −1(⊥).

Dually, it classifies open subsets in that any open subspace A is χ A −1(⊤). Note that the closed subsets and open subsets of X are related by a bijection through complementation; one gets a topology on the set of either by identifying the set with Top(X,Σ) for a suitable function space? topology.

References

  • Paul Taylor, Foundations for computable topology – 7 The Sierpinski space (html)

Revised on March 17, 2012 09:18:04 by Urs Schreiber (89.204.138.207)
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