|WikiProject Mathematics||(Rated Start-class, Mid-importance)|
Why not use ≤ for the partial order? -- Miguel
Well, I want to make the real numbers into a directed set by directing them towards a number x0, and then the notation ≤ would clash with the ordinary order of the reals. Ideally, we would have a separate symbol, such as a round less-that symbol, or a less-than symbol with a tilde underneath. AxelBoldt
I extended the page to include the notion of a directed subset, that appears quite frequently in order theory. I leaft the rest as it was. In a first change I made an error that is now corrected. -- Markus (no login)
I added the condition that a directed set should be nonempty. This is often forgotten, but should be there: many of the convenient properties of directed sets are false without it. For instance, take the statement that in a compact space, every net (= map from a directed set) has a convergent subnet. This is only true if you insist that a directed set must be nonempty. Another reason to build nonemptiness into the definition is that it makes the following true: a directed set is a preordered set A in which every finite subset has an upper bound. (This is perhaps the best definition of directed set.) By induction, this is equivalent to saying that every subset of cardinality 0 or 2 has an upper bound. The "cardinality 0" part says that A contains an element, i.e. is nonempty. The "cardinality 2" part says that for all a, b, there exists c bigger than both a and b. -- Tom (no login)
preorder and base of filters
As you allude to "(partial) orders", you could mention that a refexive and transitive relation is called a preorder.
For the type of preorder occuring here, I knew the terminology of "right filtering preorder".
Concerning limits, I knew them rather defined in terms of bases of filters instead of directed sets. Note that given a directed set A, you have canonically associated a base of filters (the set of all elements "bigger than" x, where x runs through A ; for a left filtering relation these are just the "balls of radius x"), but the converse is not true. MFH 23:15, 8 Mar 2005 (UTC)
The definition at the top says that a directed set is a "right filtering preorder". If "right filtering" is a general term, there should be a link to a definition. I'm not sure where to look for one. Also, right at the top is a description of how, starting with a and b with a ≤ b, you can create a sequence with a ≤ b ≤ c ≤ d …. Since c, d, … can all be equal to b, this statement holds for any preorder, and doesn't seem the least bit interesting. I'm removing it. Dfeuer 18:41, 2 October 2007 (UTC)