Sierpinski Space

The Sierpinski space \(S\) is a topological space consisting of two points, \(0\), and \(1\), with topology \(\mathcal{T} = \{\emptyset, 1, S\}\).

Category Theory

Both the functor which sends a topological space to its set of open sets and that the functor which sends a topological space to its set of closed sets are representable functors, and are in fact both represented by the Sierpinski space.