Well-order - Order Topology

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 points - these are the minimum and the elements with a predecessor
• limit points - 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 ω, for example N.

For subsets we can distinguish:

• Subsets with a maximum (that is, subsets which are bounded by itself); 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 which are unbounded by itself 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 which are unbounded in the whole set.

A subset is cofinal in the whole set if and only if it is unbounded in the whole set or it has a maximum which 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 ω₁ (omega-one), that is, if and only if the set is countable or has the smallest uncountable order type.