Sierpiński space

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In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński.

The Sierpiński space has important relations to the theory of computation and semantics.^[1]^[2]

Definition and fundamental properties

Explicitly, the Sierpiński space is a topological space S whose underlying point set is {0,1} and whose open sets are

$\{\varnothing ,\{1\},\{0,1\}\}.$

The closed sets are

$\{\varnothing ,\{0\},\{0,1\}\}.$

So the singleton set {0} is closed (but not open) and the set {1} is open (but not closed).

The closure operator on S is determined by

$\overline {\{0\}}=\{0\},\qquad \overline {\{1\}}=\{0,1\}.$

A finite topological space is also uniquely determined by its specialization preorder. For the Sierpiński space this preorder is actually a partial order and given by

$0\leq 0,\qquad 0\leq 1,\qquad 1\leq 1.$

Topological properties

The Sierpiński space S is a special case of both the finite particular point topology (with particular point 1) and the finite excluded point topology (with excluded point 0). Therefore S has many properties in common with one or both of these families.

Separation

The points 0 and 1 are topologically distinguishable in S since {1} is an open set which contains only one of these points. Therefore S is a Kolmogorov (T₀) space.
However, S is not T₁ since the point 1 is not closed. It follows that S is not Hausdorff, or T_n for any n ≥ 1.
S is not regular (or completely regular) since the point 1 and the disjoint closed set {0} cannot be separated by neighborhoods. (Also regularity in the presence of T₀ would imply Hausdorff.)
S is vacuously normal and completely normal since there are no nonempty separated sets.
S is not perfectly normal since the disjoint closed sets ∅ and {0} cannot be precisely separated by a function. Indeed {0} cannot be the zero set of any continuous function S → R since every such function is constant.

Connectedness

The Sierpiński space S is both hyperconnected (since every nonempty open set contains 1) and ultraconnected (since every nonempty closed set contains 0).
It follows that S is both connected and path connected.
A path from 0 to 1 in S is given by the function: f(0) = 0 and f(t) = 1 for t > 0. The function f : I → S is continuous since f⁻¹(1) = (0,1] which is open in I.
Like all finite topological spaces, S is locally path connected.
The Sierpiński space is contractible, so the fundamental group of S is trivial (as are all the higher homotopy groups).

Compactness

Like all finite topological spaces, the Sierpiński space is both compact and second-countable.
The compact subset {1} of S is not closed showing that compact subsets of T₀ spaces need not be closed.
Every open cover of S must contain S itself since S is the only open neighborhood of 0. Therefore every open cover of S has an open subcover consisting of a single set: {S}.
It follows that S is fully normal.^[3]

Convergence

Every sequence in S converges to the point 0. This is because the only neighborhood of 0 is S itself.
A sequence in S converges to 1 if and only if the sequence contains only finitely many terms equal to 0 (i.e. the sequence is eventually just 1's).
The point 1 is a cluster point of a sequence in S if and only if the sequence contains infinitely many 1's.
Examples:
- 1 is not a cluster point of (0,0,0,0,…).
- 1 is a cluster point (but not a limit) of (0,1,0,1,0,1,…).
- The sequence (1,1,1,1,…) converges to both 0 and 1.

Metrizability

The Sierpiński space S is not metrizable or even pseudometrizable since every pseudometric space is completely regular but the Sierpiński space it is not even regular.
S is generated by the hemimetric (or pseudo-quasimetric) $d(0,1)=0$ and $d(1,0)=1$ .

Other properties

There are only three continuous maps from S to itself: the identity map and the constant maps to 0 and 1.
It follows that the homeomorphism group of S is trivial.

Continuous functions to the Sierpiński space

Let X be an arbitrary set. The set of all functions from X to the set {0,1} is typically denoted 2^X. These functions are precisely the characteristic functions of X. Each such function is of the form

$\chi _{U}(x)={\begin{cases}1&x\in U\\0&x\not \in U\end{cases}}$

where U is a subset of X. In other words, the set of functions 2^X is in bijective correspondence with P(X), the power set of X. Every subset U of X has its characteristic function χ_U and every function from X to {0,1} is of this form.

Now suppose X is a topological space and let {0,1} have the Sierpiński topology. Then a function χ_U : X → S is continuous if and only if χ_U⁻¹(1) is open in X. But, by definition

$\chi _{U}^{{-1}}(1)=U.$

So χ_U is continuous if and only if U is open in X. Let C(X,S) denote the set of all continuous maps from X to S and let T(X) denote the topology of X (i.e. the family of all open sets). Then we have a bijection from T(X) to C(X,S) which sends the open set U to χ_U.

$C(X,S)\cong {\mathcal {T}}(X)$

That is, if we identify 2^X with P(X), the subset of continuous maps C(X,S) ⊂ 2^X is precisely the topology of X: T(X) ⊂ P(X).

Categorical description

The above construction can be described nicely using the language of category theory. There is contravariant functor T : Top → Set from the category of topological spaces to the category of sets which assigns each topological space X its set of open sets T(X) and each continuous function f : X → Y the preimage map

$f^{{-1}}:{\mathcal {T}}(Y)\to {\mathcal {T}}(X).$

The statement then becomes: the functor T is represented by (S, {1}) where S is the Sierpiński space. That is, T is naturally isomorphic to the Hom functor Hom(–, S) with the natural isomorphism determined by the universal element {1} ∈ T(S).

The initial topology

Any topological space X has the initial topology induced by the family C(X,S) of continuous functions to Sierpiński space. Indeed, in order to coarsen the topology on X one must remove open sets. But removing the open set U would render χ_U discontinuous. So X has the coarsest topology for which each function in C(X,S) is continuous.

The family of functions C(X,S) separates points in X if and only if X is a T₀ space. Two points x and y will be separated by the function χ_U if and only if the open set U contains precisely one of the two points. This is exactly what it means for x and y to be topologically distinguishable.

Therefore if X is T₀, we can embed X as a subspace of a product of Sierpiński spaces, where there is one copy of S for each open set U in X. The embedding map

$e:X\to \prod _{{U\in {\mathcal {T}}(X)}}S=S^{{{\mathcal {T}}(X)}}$

is given by

$e(x)_{U}=\chi _{U}(x).\,$

Since subspaces and products of T₀ spaces are T₀, it follows that a topological space is T₀ if and only if it is homeomorphic to a subspace of a power of S.

In algebraic geometry

In algebraic geometry the Sierpiński space arises as the spectrum, Spec(R), of a discrete valuation ring R such as Z₍₂₎ (the localization of the integers at the prime ideal generated by 2). The generic point of Spec(R), coming from the zero ideal, corresponds to the open point 1, while the special point of Spec(R), coming from the unique maximal ideal, corresponds to the closed point 0.

Notes

↑ An online paper, it explains the motivation, why the notion of “topology” can be applied in the investigation of concepts of the computer science. Alex Simpson: Mathematical Structures for Semantics. Chapter III: Topological Spaces from a Computational Perspective. The “References” section provides many online materials on domain theory.
↑ Escardó, Martín (2004). Synthetic topology of data types and classical spaces. Electronic Notes in Theoretical Computer Science 87. Elsevier. Retrieved July 6, 2011.
↑ Steen and Seebach incorrectly list the Sierpiński space as not being fully normal (or fully T₄ in their terminology).

References

Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 507446

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