Two point topological space

Question

Is there a standard name for the two point space with precisely one singleton being the only nontrivial open set?

What are its most noteworthy categorical properties?

Answer 1

Yes! It is the Sierpinski Space. You will find most answers in the wikipedia link. It is a connected two point set and is really useful for plenty counterexamples and/or constructions.

From a categorical viewpoint the Sierpinski Space represents the functor $X \mapsto \tau (X)$, $f\mapsto f^{-1}$.

Answer 2

The Sierpinski space is a dualizing object which mediates the Stone duality between the category of topological spaces and the category of frames.

http://ncatlab.org/nlab/show/dualizing+object