up:: Topological Space
The Sierpiński Space is defined over a set with two elements, with the Topology that only one of those is an open set (the other, thus, is closed).
Formally, it is the set
Properties
Through the Sierpiński space, one can prove that Every topology has a bijection with the Hom-Set of its continuous functions to the Sierpiński space: that is, given a topology, each open set has an associated continuous function which “picks it out” from the Sierpiński’s space non-trivial open set.