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 {0, 1} and whose open sets are The closed sets are So the singleton set {0} is closed and the set {1} 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 S 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, S has many properties in common with one or both of these families.

Separation

Connectedness

Compactness

Convergence

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 {0, 1} is typically denoted 2^X. 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 2^X is in bijective correspondence with P(X), the power set of X. Every subset U of X has its characteristic function \chi_U and every function from X to {0, 1} is of this form. Now suppose X is a topological space and let {0, 1} have the Sierpiński topology. Then a function is continuous if and only if is open in X. But, by definition So \chi_U is continuous if and only if U is open in X. Let C(X, S) denote the set of all continuous maps from X to S and let T(X) denote the topology of X (that is, the family of all open sets). Then we have a bijection from T(X) to C(X, S) which sends the open set U to \chi_U. That is, if we identify 2^X with P(X) the subset of continuous maps is precisely the topology of X: 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 X its set of open sets T(X) and each continuous function f : X \to Y the preimage map The statement then becomes: the functor T is represented by (S, {1}) where S is the Sierpiński space. That is, T 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 C(X, S) 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 \chi_U discontinuous. So X has the coarsest topology for which each function in C(X, S) is continuous. The family of functions C(X, S) separates points in X if and only if X is a T0 space. Two points x and y will be separated by the function \chi_U if and only if the open set U contains precisely one of the two points. This is exactly what it means for x and y 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 R such as \Z_{(p)} (the localization of the integers at the prime ideal generated by the prime number p). 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.