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Sierpinski space

In topology, Sierpinski space S is the simplest example of a topological space that does not satisfy the T1 axiom. It is useful as a counterexample and has many interesting properties related to general topological considerations.

Definition   Let S = {0,1}. Then T = is a topology on S, and the resulting topological space is called Sierpinski space.

Useful Facts

The Sierpinski space S has several interesting properties.

The Sierpinski space has important relations to the theory of computation and semantics. See Alex Simpson lectures for Mathematical Structures for Semantics