Editing Well-order (section)

== Order topology==
Every well-ordered set can be made into a [[topological space]] by endowing it with the [[order topology]].

With respect to this topology there can be two kinds of elements:
*[[isolated point]]s — these are the minimum and the elements with a predecessor.
*[[limit point]]s — this type does not occur in finite sets, and may or may not occur in an infinite set; the infinite sets without limit point are the sets of order type {{mvar|ω}}, for example the [[natural numbers]] {{tmath|\N.}}

For subsets we can distinguish:
*Subsets with a maximum (that is, subsets that are [[Bounded set#Boundedness in order theory|bounded]] by themselves); this can be an isolated point or a limit point of the whole set; in the latter case it may or may not be also a limit point of the subset.
*Subsets that are unbounded by themselves but bounded in the whole set; they have no maximum, but a supremum outside the subset; if the subset is non-empty this supremum is a limit point of the subset and hence also of the whole set; if the subset is empty this supremum is the minimum of the whole set.
*Subsets that are unbounded in the whole set.

A subset is [[Cofinal (mathematics)|cofinal]] in the whole set if and only if it is unbounded in the whole set or it has a maximum that is also maximum of the whole set.

A well-ordered set as topological space is a [[first-countable space]] if and only if it has order type less than or equal to ω<sub>1</sub> ([[first uncountable ordinal|omega-one]]), that is, if and only if the set is [[countable]] or has the smallest [[uncountable]] order type.