In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Sierpiński space (or the connected two-point set) is a
finite topological space
In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements.
Finite topological spaces are often used to provide example ...
with two points, only one of which is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
.
It is the smallest example of a
topological space which is neither
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
nor
discrete. It is named after
Wacław Sierpiński
Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and to ...
.
The Sierpiński space has important relations to the
theory of computation and
semantics, 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
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
is a special case of both the finite
particular point topology In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collect ...
(with particular point 1) and the finite
excluded point topology In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collection
:T = \ \cup \
of subsets of ''X'' is then the excluded ...
(with excluded point 0). Therefore,
has many properties in common with one or both of these families.
Separation
*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 (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
*''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 set
In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets ...
s.
*''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
*The Sierpiński space ''S'' is both
hyperconnected
In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is pre ...
(since every nonempty open set contains 1) and
ultraconnected
In mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint.PlanetMath Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersectio ...
(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
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
(as are all the
higher homotopy groups).
Compactness
*Like all finite topological spaces, the Sierpiński space is both
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
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
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alp ...
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.
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
**1 is a cluster point (but not a limit) of
**The sequence
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 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
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
and the
constant map
In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image).
Basic properti ...
s to 0 and 1.
*It follows that the
homeomorphism group In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important ...
of ''S'' is
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or fork ...
.
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
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
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 set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
s, in which the Sierpiński space becomes the
classifying space for open sets when the characteristic function preserves
directed joins.
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
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
to the
Hom functor with the natural isomorphism determined by the
universal element This is generalized by the notion of a
presheaf.
[Saunders MacLane, Ieke Moerdijk, ''Sheaves in Geometry and Logic: A First Introduction to Topos Theory'', (1992) Springer-Verlag Universitext ]
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 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,
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
*
*
*
Notes
References
*
* Michael Tiefenback (1977) "Topological Genealogy",
Mathematics Magazine 50(3): 158–60
{{DEFAULTSORT:Sierpinski space
General topology
Topological spaces