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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 = },{1},{0,1}} 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

General topology

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Topics: Sierpinski Space, Sierpinski Carpet, Sierpinski Triangle...

Sierpinski carpetFractals L-system of six iterations. Iterated function system. The Sierpinski carpet named after Waclaw Sierpinski, is a fractal derived from a square by cutting it into 9 equal squares with a 3-by-3 grid, removing the central piece and then applying the	Sierpinski triangleThe Sierpinski triangle also called the Sierpinski gasket is a fractal, named after Waclaw Sierpinski. An algorithm for obtaining arbitrarily close approximations to the Sierpinski triangle is as follows: #Start with any triangle in a plane. The canonical	Sierpinski numberAnalytic number theory In mathematics, a Sierpinski number is a positive, odd, number k such that integers of the form k''2''n + 1 are composite (i. not prime) for all natural numbers n''. In other words, when k is a Sierpinski number, all members of the
Sierpinski spaceIn topology, Sierpinski space S is the simplest example of a topological space that does not satisfy the T axiom. It is useful as a counterexample and has many interesting properties related to general topological considerations. Definition Let S {0,1}.	Sierpinski curveA curve which describes a closed path which contains every interior point of a square. The length of the curve is infinity, while the area enclosed by it is 5/12 that of the square. The mathematics for this curve were pioneered by Waclaw Sierpinski.

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