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.
Explicitly, the Sierpiński space is a topological space S whose underlying point set is
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
has many properties in common with one or both of these families.
In other words, the set of functions
Every subset U of X has its characteristic function
Now suppose X is a topological space and 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).
the subset of continuous maps
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] The above construction can be described nicely using the language of category theory.
from the category of topological spaces to the category of sets which assigns each topological space
is naturally isomorphic to the Hom functor
with the natural isomorphism determined by the universal element
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
So X has the coarsest topology for which each function in
separates points in X if and only if X is a T0 space.
if and only if the open set U contains precisely one of the two points.
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.
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 the Sierpiński space arises as the spectrum
of a discrete valuation ring
coming from the unique maximal ideal, corresponds to the closed point 0.