Every dcpo is an ?-cpo, since every ?-chain is a directed set, but the converse is not true.
22.
Starting with an arbitrary set, you want to use the existence of finite joins to turn it into a directed set, as you thought.
23.
The requirement that the metric take values in [ 0, " ) can even be relaxed to consider metrics with values in other directed sets.
24.
Is there some reasonable example of a directed set that is not a poset ?-talk ) 02 : 06, 4 November 2008 ( UTC)
25.
Indeed, in the theory of metric spaces, sequences play a role that is in many aspects analogous to the role of directed sets in domain theory.
26.
The compact elements of a domain have the important special property that they cannot be obtained as a limit of a directed set in which they did not already occur.
27.
This leads to the class of continuous posets, consisting of posets where every element can be obtained as the supremum of a directed set of elements that are algebraic.
28.
It was established by an Act of the Virginia House of Burgesses in 1691, when each county in the Virginia Colony was directed set aside of land for a town.
29.
Requiring the existence of directed suprema can be motivated by viewing directed sets as generalized approximation sequences and suprema as " limits " of the respective ( approximative ) computations.
30.
It seems to me that the collection of all finite cyclic groups also forms a direct system of groups over the directed set of natural numbers ordered ( dually ) by divisibility.
