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We study the Sierpinski object $\Sigma$ in the realizability topos based onScott's graph model of the $\lambda$-calculus. Our starting observation is thatthe object of realizers in this topos is the exponential $\Sigma ^N$, where $N$is the natural numbers object. We define order-discrete objects byorthogonality to $\Sigma$. We show that the order-discrete objects form areflective subcategory of the topos, and that many fundamental objects inhigher-type arithmetic are order-discrete. Building on work by Lietz, we givesome new results regarding the internal logic of the topos. Then we consider$\Sigma$ as a dominance; we explicitly construct the lift functor andcharacterize $\Sigma$-subobjects. Contrary to our expectations the dominance$\Sigma$ is not closed under unions. In the last section we build a model forhomotopy theory, where the order-discrete objects are exactly those objectswhich only have constant paths.

The Sierpinski Object in the Scott Realizability Topos