Сиерпински кеңістігі: математикалық анықтама және қасиеттері

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.

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.

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.

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).

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.

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.

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.

In algebraic geometry the Sierpiński space arises as the spectrum of a discrete valuation ring such as (the localization of the integers at the prime ideal generated by the prime number ). 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.