nLab
Sierpinski space

Contents

Definition

Definition

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

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 closed subspaces

The Sierpinski space is a classifier for closed subspaces of a topological space XX in that for any closed 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}(\bot).

Dually, it classifies open subsets in that any open subspace AA is χ A 1()\chi_A^{-1}(\top). 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.

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)