Sierpinski space
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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.

• S is an inaccessible Kolmogorov space; i.e. S satisfies the T₀ axiom, but not the T₁ axiom.
• A topological space is Kolmogorov if and only if it is homeomorphic to a subspace of a power of S.
• For any topological space X with topology T, let C(X,S) denote the set of all continuous maps from X to S, and for each subset A of X, let I(A) denote the indicator function of A. Then the mapping f : T → C(X,S) defined by f(U) = I(U) is a bijective correspondence.
• If X is a topological space with topology T, then the weak topology on X generated by C(X,S) coincides with T.

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

The original article can be found at: http://en.wikipedia.org/wiki/Sierpinski_space.
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