In mathematics, the Sierpiński space (or the connected two-point set) 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 ( 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
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 Tn for any S is not regular (or completely regular) since the point 1 and the disjoint closed set cannot be separated by neighborhoods. (Also regularity in the presence of T0 would imply Hausdorff.) 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 (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.
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
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 (or pseudo-quasimetric) 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 (that is, the family of all open sets). Then we have a bijection from to which sends the open set to