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
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![{\displaystyle {\overline {\{0\}}}=\{0\},\qquad {\overline {\{1\}}}=\{0,1\}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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![{\displaystyle 0\leq 0,\qquad 0\leq 1,\qquad 1\leq 1.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Topological properties
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
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:
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
- 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:
![{\displaystyle \{S\}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- It follows that S is fully normal.[4]
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
![{\displaystyle (0,0,0,0,\ldots ).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- 1 is a cluster point (but not a limit) of
![{\displaystyle (0,1,0,1,0,1,\ldots ).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- The sequence
converges to both 0 and 1.
Metrizability
Other properties
Continuous functions to the Sierpiński space
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 ![{\displaystyle \chi _{U}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
That is, if we identify
with
the subset of continuous maps
is precisely the topology of
![{\displaystyle T(X)\subseteq P(X).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
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
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
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
- Finite topological space – topological space with a finite number of pointsPages displaying wikidata descriptions as a fallback
- 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
- ^ Sierpinski space at the nLab
- ^ 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 T4 in their terminology).
- ^ Scott topology at the nLab
- ^ Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, (1992) Springer-Verlag Universitext ISBN 978-0387977102
References