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.
The Sierpiński space has the set of points
Its open sets are
Thus, the only nontrivial open set is . It can be thought of as an abstract "observation of termination".
The above topology is also induced as the Scott topology of the flat cpo
The following is a classic lemma in point-set topology:
Lemma. Let be a space. Then open sets of correspond to continuous functions .
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