# Sierpiński space

In mathematics, the **Sierpiński space** is a finite topological space with two points, only one of which is closed.^{[1]}
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,^{[2]}^{[3]} because it is the classifying space for open sets in the Scott topology.

## Definition and fundamental properties[edit]

Explicitly, the Sierpiński space is a topological space *S* whose underlying point set is and whose open sets are
The closed sets are
So the singleton set is closed and the set is open ( is the empty set).

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[edit]

The Sierpiński space 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, has many properties in common with one or both of these families.

### Separation[edit]

- The points 0 and 1 are topologically distinguishable in
*S*since 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 *S*is not regular (or completely regular) since the point 1 and the disjoint closed set 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 cannot be precisely separated by a function. Indeed, cannot be the zero set of any continuous function since every such function is constant.

### Connectedness[edit]

- 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: and for The function is continuous since 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[edit]

- Like all finite topological spaces, the Sierpiński space is both compact and second-countable.
- The compact subset 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: - It follows that
*S*is fully normal.^{[4]}

### Convergence[edit]

- 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
- 1 is a cluster point (but not a limit) of
- The sequence converges to both 0 and 1.

### Metrizability[edit]

- The Sierpiński space
*S*is not metrizable or even pseudometrizable since every pseudometric space is completely regular but the Sierpiński space is not even regular. *S*is generated by the hemimetric (or pseudo-quasimetric) and

### Other properties[edit]

- 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[edit]

Let *X* be an arbitrary set. The set of all functions from *X* to the set is typically denoted 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 is in bijective correspondence with the power set of *X*. Every subset *U* of *X* has its characteristic function and every function from *X* to is of this form.

Now suppose *X* is a topological space and let have the Sierpiński topology. Then a function is continuous if and only if is open in *X*. But, by definition
So is continuous if and only if *U* is open in *X*. Let denote the set of all continuous maps from *X* to *S* and let denote the topology of *X* (that is, the family of all open sets). Then we have a bijection from to which sends the open set to
That is, if we identify with the subset of continuous maps is precisely the topology of

A particularly notable example of this is the Scott topology for partially ordered sets, in which the Sierpiński space becomes the classifying space for open sets when the characteristic function preserves directed joins.^{[5]}

### Categorical description[edit]

The above construction can be described nicely using the language of category theory. There is a contravariant functor from the category of topological spaces to the category of sets which assigns each topological space its set of open sets and each continuous function the preimage map
The statement then becomes: the functor is represented by where is the Sierpiński space. That is, is naturally isomorphic to the Hom functor with the natural isomorphism determined by the universal element This is generalized by the notion of a presheaf.^{[6]}

### The initial topology[edit]

Any topological space *X* has the initial topology induced by the family 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 discontinuous. So *X* has the coarsest topology for which each function in is continuous.

The family of functions separates points in *X* if and only if *X* is a T_{0} space. Two points and will be separated by the function if and only if the open set *U* contains precisely one of the two points. This is exactly what it means for and 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[edit]

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

## See also[edit]

- Finite topological space – topological space with a finite number of points
- Freyd cover, a categorical construction related to the Sierpiński space
- List of topologies – List of concrete topologies and topological spaces
- Pseudocircle – Four-point non-Hausdorff topological space

## Notes[edit]

**^**Sierpinski space at the*n*Lab**^**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 (original). Chapter III: Topological Spaces from a Computational Perspective (original). 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. Vol. 87. Elsevier. p. 2004. CiteSeerX 10.1.1.129.2886.**^**Steen and Seebach incorrectly list the Sierpiński space as*not*being fully normal (or fully T_{4}in their terminology).**^**Scott topology at the*n*Lab**^**Saunders MacLane, Ieke Moerdijk,*Sheaves in Geometry and Logic: A First Introduction to Topos Theory*, (1992) Springer-Verlag Universitext ISBN 978-0387977102

## References[edit]

- 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 0507446 - Michael Tiefenback (1977) "Topological Genealogy", Mathematics Magazine 50(3): 158–60 doi:10.2307/2689505