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
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 (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:
- 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
- 1 is a cluster point (but not a limit) of
- 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
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
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 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