"greater than or equal to" (≥) relations is totally ordered. Hence each subset of the real numbers is totally ordered, such as the natural numbers, integers, and rational numbers. Each of these can be shown to be the unique (up to an order isomorphism) "initial example" of a totally ordered set with a certain property, (here, a total order is initial for a property, if, whenever has the property, there is an order isomorphism from to a subset of ): The natural numbers form an initial non-empty totally ordered set with no upper bound. The integers form an initial non-empty totally ordered set with neither an upper nor a lower bound. The rational numbers form an initial totally ordered set which is dense in the real numbers. Moreover, the reflexive reduction < is a dense order on the rational numbers. The real numbers form an initial unbounded totally ordered set that is connected in the order topology (defined below). Ordered fields are totally ordered by definition. They include the rational numbers and the real numbers. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. Any Dedekind-complete ordered field is isomorphic to the real numbers. The letters of the alphabet ordered by the standard dictionary order, e.g., etc., is a strict total order. Chains The term chain is sometimes defined as a synonym for a totally ordered set, but it is generally used for referring to a subset of a partially ordered set that is totally ordered for the induced order. Typically, the partially ordered set is a set of subsets of a given set that is ordered by inclusion, and the term is used for stating properties of the set of the chains. This high number of nested levels of sets explains the usefulness of the term. A common example of the use of chain for referring to totally ordered subsets is Zorn's lemma which asserts that, if every chain in a partially ordered set has an upper bound in , then contains at least one maximal element. Zorn's lemma is commonly used with being a set of subsets; in this case, the upperbound is obtained by proving that the union of the elements of a chain in is in . This is the way that is generally used to prove that a vector space has Hamel bases and that a ring has maximal ideals.