# 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

The closed sets are

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

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

## 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 (T_{0}) space. - However,
*S*is not T_{1}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 T_{0}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 T_{0}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) and .

### 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

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

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}.

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

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 T_{0} 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 T_{0}, 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

is given by

Since subspaces and products of T_{0} spaces are T_{0}, it follows that a topological space is T_{0} 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.

## See also

## 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.
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ Steen and Seebach incorrectly list the Sierpiński space as
*not*being fully normal (or fully T_{4}in their terminology).

## References

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