# nLab Sierpinski space

### Context

#### Topology

topology

algebraic topology

# Contents

## Definition

###### Definition

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

###### Remark

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

### 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 ${\chi }_{A}:X\to S$ such that $A={\chi }_{A}^{-1}\left(\perp \right)$.

Dually, it classifies open subsets in that any open subspace $A$ is ${\chi }_{A}^{-1}\left(\top \right)$. 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 $Top\left(X,\Sigma \right)$ 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)