| 0 ≤ n < ω } has order type ω. The set of numbers { − 2−n − 2−m−n | 0 ≤ m,n < ω } has order type ω2. The previous set is the set of limit points within the set. Within the set of real numbers, either with the ordinary topology or the order topology, 0 is also a limit point of the set. It is also a limit point of the set of limit points. The set of numbers { − 2−n | 0 ≤ n < ω } ∪ { 1 } has order type ω + 1. With the order topology of this set, 1 is a limit point of the set. With the ordinary topology (or equivalently, the order topology) of the real numbers it is not. Equivalent formulations If a set is totally ordered, then the following are equivalent to each other: The set is well ordered. That is, every nonempty subset has a least element. Transfinite induction works for the entire ordered set. Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming the axiom of dependent choice). Every subordering is isomorphic to an initial segment. 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 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 which 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 which are unbounded in the whole set. A subset is cofinal