Sierpinski space
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In topology, the Sierpiński space is the simplest non-trivial, non-discrete topological space. It is also 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 <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.
The Sierpiński space has important relations to the theory of computation and semantics. See Alex Simpson lectures for Mathematical Structures for Semantics (http://www.dcs.ed.ac.uk/home/als/Teaching/MSfS/).