Sierpinski space

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In mathematics, Sierpiński space is a simple topological space named after Wacław Sierpiński. Although it has only two points, it is often used as a counterexample in topology and has some interesting properties related to general topological considerations.

Definition

Let <math>S = \{0,1 \}</math>. Then <math>T = \{\varnothing, \{1 \}, \{0,1 \} \}</math> is a topology on S, and the resulting topological space is called Sierpiński space.

Useful facts

The Sierpiński 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.
• S is a sober space.

The Sierpiński space has important relations to the theory of computation and semantics. [1].

References

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition). (See example 11)