The Sierpiński space is the topological space which is the set of truth values, classically , equipped with the specialization topology, in which is closed and is open but not conversely. (The opposite convention is also used.)
In constructive mathematics, it is important that be open (and closed), rather than the other way around. Indeed, the general definition (since we can't assume that every element is either or ) is that a subset of is open as long as it is upward closed: and imply that . The ability to place a topology on is fundamental to abstract Stone duality, a constructive approach to general topology.
This Sierpinski space
Dually, it classifies open subsets in that any open subspace is . Note that the closed subsets and open subsets of are related by a bijection through complementation; one gets a topology on the set of either by identifying the set with for a suitable function space? topology.