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

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 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:  
  • 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

Other properties edit

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 T0 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 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

 
is given by
 
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 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

Notes edit

  1. ^ Sierpinski space at the nLab
  2. ^ 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.
  3. ^ 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.
  4. ^ Steen and Seebach incorrectly list the Sierpiński space as not being fully normal (or fully T4 in their terminology).
  5. ^ Scott topology at the nLab
  6. ^ Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, (1992) Springer-Verlag Universitext ISBN 978-0387977102

References edit