# Two point topological space

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?

• Just to symbolize: $(X,\tau) = (\{1,2\}, \left\{\{1,2\}, \{1\}, \emptyset\right\})$? Sep 23, 2014 at 10:32
• @AlexR yes exactly Sep 23, 2014 at 10:36

From a categorical viewpoint the Sierpinski Space represents the functor $X \mapsto \tau (X)$, $f\mapsto f^{-1}$.
• The topology of the Sierpinski space is the Scott topology of the poset $\{0, 1\}$ (partially ordered by $0 \leq 1$). Sep 23, 2014 at 11:45