Sierpiński space

In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed.
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, 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.
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 total order and given by

Topological properties

The Sierpiński space is a special case of both the finite particular point topology and the finite excluded point topology. 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 since the point 1 and the disjoint closed set cannot be separated by neighborhoods.
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

The Sierpiński space S is both hyperconnected and ultraconnected.
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.

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.

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.
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 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 and

Other properties

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

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. 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.

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.

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 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.