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.

• 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