TITLE: Two point topological space
QUESTION [5 upvotes]: 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?

REPLY [2 votes]: 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