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 areThe 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 partial order and given by
Topological properties
The Sierpiński space S is a special case of both the finite particular point topology and the finite excluded point topology. Therefore, S 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 space.
- However, S is not T1 since the point 1 is not closed. It follows that S is not Hausdorff, or Tn for any n ≥ 1.
- 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 S → R 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: f = 0 and f = 1 for t > 0. The function f : I → S is continuous since f−1 =.
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
where U is a subset of X. In other words, the set of functions 2X is in bijective correspondence with P, the power set of X. Every subset U of X has its characteristic function χU 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 χU : X → S is continuous if and only if χU−1 is open in X. But, by definition
So χU is continuous if and only if U is open in X. Let C denote the set of all continuous maps from X to S and let T denote the topology of X. Then we have a bijection from T to C which sends the open set U to χU.
That is, if we identify 2X with P, the subset of continuous maps C ⊂ 2X is precisely the topology of X: T ⊂ P.
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 contravariant functor T : Top → Set from the category of topological spaces to the category of sets which assigns each topological space X its set of open sets T and each continuous function f : X → Y the preimage mapThe statement then becomes: the functor T is represented by where S is the Sierpiński space. That is, T is naturally isomorphic to the Hom functor Hom with the natural isomorphism determined by the universal element ∈ T. 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 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 χU discontinuous. So X has the coarsest topology for which each function in C is continuous.The family of functions C separates points in X if and only if X is a T0 space. Two points x and y will be separated by the function χU 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.