nLab
Sierpinski space

Definition

Definition

The Sierpiński space $\Sigma$ is the topological space

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

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

(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 $P$ of $\Sigma$ is open as long as it is upward closed: $p \Rightarrow q$ and $p \in P$ imply that $q \in P$. The ability to place a topology on $\Top(X,\Sigma)$ is fundamental to abstract Stone duality, a constructive approach to general topology.

Properties

As a topological space

This Sierpinski space

• is a contractible space;

• is a locally contractible space;

• has a focal point;

• is a sober topological space (but not T1).

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

As a classifer for open/closed subspaces

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

Dually, it classifies closed subsets in that any closed subspace $A$ is $\chi_A^{-1}(\bot)$. 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 $\Top(X,\Sigma)$ for a suitable function space topology. (This part does not work as well in constructive mathematics.)

Related concepts

• empty space, point space

• Cantor space

• Sierpinski topos, Sierpinski (∞,1)-topos

• Stone duality

• In synthetic topology, an analogue of the Sierpinski space is called a dominance.

References

• Wikipedia, Sierpinski space

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