Sierpiński space

The Sierpiński space is a two-point topological space. It plays a particularly important role in the domain-theoretic approach to semantics of programming languages.

Definition

Point-set definition

The Sierpiński space has the set of points

\Sigma \triangleq \{ \bot, \top \}

Its open sets are

  \{ \emptyset, \{ \top \}, \{ \bot, \top \} \}

Thus, the only nontrivial open set is \{ \top \}. It can be thought of as an abstract "observation of termination".

Through orders

The above topology is also induced as the Scott topology of the flat cpo

\Sigma \triangleq \{ \bot, \top \},
\quad
\bot \sqsubseteq \top

Properties

The following is a classic lemma in point-set topology:

Lemma. Let X be a space. Then open sets of X correspond to continuous functions X \to \Sigma.

References

• Wikipedia page

• nLab page

• A discussion of the Sierpiński space by Paul Taylor in his book on Abstract Stone Duality. Note the quote:

The Sierpiński space was treated with derision in classical topology, but it is the spider in the middle of the web in our subject