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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?

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

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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 $X \mapsto \tau (X)$, $f\mapsto f^{-1}$.

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    $\begingroup$ The topology of the Sierpinski space is the Scott topology of the poset $\{0, 1\}$ (partially ordered by $0 \leq 1$). $\endgroup$
    – polmath
    Sep 23, 2014 at 11:45
  • $\begingroup$ I knew about the the representation. Any other interesting categorical property? $\endgroup$
    – magma
    Sep 23, 2014 at 12:52
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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

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