Топологиялық кеңістіктегі мамандық алдынан реттелуі

In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space. For most spaces that are considered in practice, namely for all those that satisfy the T0 separation axiom, this preorder is even a partial order (called the specialization order). On the other hand, for T1 spaces the order becomes trivial and is of little interest. The specialization order is often considered in applications in computer science, where T0 spaces occur in denotational semantics. The specialization order is also important for identifying suitable topologies on partially ordered sets, as is done in order theory.

Let X be a topological space and let ≤ be the specialization preorder on X. Every open set is an upper set with respect to ≤ and every closed set is a lower set. The converses are not generally true. In fact, a topological space is an Alexandrov discrete space if and only if every upper set is also open (or equivalently every lower set is also closed). Let A be a subset of X. The smallest upper set containing A is denoted ↑A
and the smallest lower set containing A is denoted ↓A. In case A = {x} is a singleton one uses the notation ↑x and ↓x. For x ∈ X one has:

↑x = {y ∈ X : x ≤ y} = ∩{open sets containing x}. ↓x = {y ∈ X : y ≤ x} = ∩{closed sets containing x} = cl{x}. The lower set ↓x is always closed; however, the upper set ↑x need not be open or closed. The closed points of a topological space X are precisely the minimal elements of X with respect to ≤.

In the Sierpinski space {0,1} with open sets {∅, {1}, {0,1 the specialization order is the natural one (0 ≤ 0, 0 ≤ 1, and 1 ≤ 1). If p, q are elements of Spec(R) (the spectrum of a commutative ring R) then p ≤ q if and only if q ⊆ p (as prime ideals). Thus the closed points of Spec(R) are precisely the maximal ideals.

The specialization order yields a tool to obtain a preorder from every topology. It is natural to ask for the converse too: Is every preorder obtained as a specialization preorder of some topology? Indeed, the answer to this question is positive and there are in general many topologies on a set X that induce a given order ≤ as their specialization order. The Alexandroff topology of the order ≤ plays a special role: it is the finest topology that induces ≤. The other extreme, the coarsest topology that induces ≤, is the upper topology, the least topology within which all complements of sets ↓x (for some x in X) are open. There are also interesting topologies in between these two extremes. The finest sober topology that is order consistent in the above sense for a given order ≤ is the Scott topology. The upper topology however is still the coarsest sober order consistent topology. In fact, its open sets are even inaccessible by any suprema. Hence any sober space with specialization order ≤ is finer than the upper topology and coarser than the Scott topology. Yet, such a space may fail to exist, that is, there exist partial orders for which there is no sober order consistent topology. Especially, the Scott topology is not necessarily sober.