Sierpinski Space

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

Useful facts

• 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.

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