Main Page | See live article | Alphabetical index 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 This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.