nLab Sierpinski space

This is about the topological space of the set of truth values equipped with the Scott topology. For the set of Sierpinski semi-decidable truth values, which is sometimes called Sierpiński space in predicative constructive mathematics, see at semi-decidable proposition.

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

The corresponding locale is given by the three-element frame $\{\bot \lt \omega \lt \top\}$.

Remark

In constructive mathematics, the Sierpiński locale is given by the free frame over one element. More concretely, its opens are $O_{p \le q}$ indexed by pairs of truth values $(p, q)$ such that $p \le q$. The partial order is given by $O_{p \le q} \le O_{p' \le q'}$ iff $p \le p'$ and $q \le q'$. This is also the Alexandroff locale over the poset of classical truth values.

The corresponding topological space however, has the point set given by all truth values. The open subsets are $O_{p \le q} = \{\varphi \mid p \vee (q \wedge \varphi)\}$. Importantly, it is not homeomorphic to the Alexandroff topology on the set of truth values, or classical truth values. It is however homeomorphic to the Scott topology on the set of truth values.

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 classifier 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

• separation axioms in terms of lifting properties

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

  • Sierpinski dominance

• Phoa's principle

References

• Wikipedia, Sierpinski space

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

• Andrej Bauer, Davorin Lešnik, Metric spaces in synthetic topology Annals of Pure and Applied Logic, Volume 163, Issue 2, February 2012, Pages 87-100 (doi:10.1016/j.apal.2011.06.017)

• Ayberk Tosun, Constructive and Predicative Locale Theory in Univalent Foundations (arXiv:2603.01308)

• Constructive subtleties about the Sierpinski Space, Mathematics StackExchange. (web)