Topologies On Orders
The specialization order yields a tool to obtain a partial order from every topology. It is natural to ask for the converse too: Is every partial order obtained as a specialization order of some topology?
Indeed, the answer to this question is positive and there are in general many topologies on a set X which 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 {y in X | y ≤ x} (for some x in X) are open.
There are also interesting topologies in between these two extremes. The finest 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 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. Especially, the Scott topology is not necessarily sober.
Read more about this topic: Specialization (pre)order
Famous quotes containing the word orders:
“There is nothing on earth more exquisite than a bonny book, with well-placed columns of rich black writing in beautiful borders, and illuminated pictures cunningly inset. But nowadays, instead of looking at books, people read them. A book might as well be one of those orders for bacon and bran.”
—George Bernard Shaw (1856–1950)