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Way-below relation

A more elaborate approach leads to the definition of the so-called order of approximation, which is more suggestively also called the way-below relation. An element x is way below an element y, if, for every directed set D with supremum such that

there is some element d in D such that

Then one also says that x approximates y and writes

This does imply that

since the singleton set { y } is directed. For an example, in an ordering of sets, an infinite set is way above any of its finite subsets. On the other hand, consider the directed set (in fact, the chain) of finite sets

Since the supremum of this chain is the set of all natural numbers N, this shows that no infinite set is way below N.

However, being way below some element is a relative notion and does not reveal much about an element alone. For example, one would like to characterize finite sets in an order-theoretic way, but even infinite sets can be way below some other set. The special property of these finite elements x is that they are way below themselves, i.e.

An element with this property is also called compact. Yet, such elements do not have to be "finite" nor "compact" in any other mathematical usage of the terms. The notation is nonetheless motivated by certain parallels to the respective notions in set theory and topology. 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.

Many other important results about the way-below relation support the claim that this definition is appropriate to capture many important aspects of a domain.