Sierpiński space

In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński.

The Sierpiński space has important relations to the theory of computation and semantics, because it is the classifying space for open sets in the Scott topology.

Explicitly, the Sierpiński space is a topological space S whose underlying point set is {0,1} and whose open sets are

The closed sets are

So the singleton set {0} is closed and the set {1} is open.

The closure operator on S is determined by

A finite topological space is also uniquely determined by its specialization preorder. For the Sierpiński space this preorder is actually a partial order and given by

The Sierpiński space S is a special case of both the finite particular point topology (with particular point 1) and the finite excluded point topology (with excluded point 0). Therefore, S has many properties in common with one or both of these families.

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