assumptions suffice to obtain such a kind of converse. For example if is a map on a T1 topological space with a unique fixed point a, such that for each we have fn(x) → a, then there already exists a metric on X with respect to which f satisfies the conditions of the Banach contraction principle with contraction constant 1/2. In this case the metric is in fact an ultrametric. Generalizations There are a number of generalizations (some of which are immediate corollaries). Let T : X → X be a map on a complete non-empty metric space. Then, for example, some generalizations of the Banach fixed-point theorem are: Assume that some iterate Tn of T is a contraction. Then T has a unique fixed point. Assume that for each n, there exist cn such that d(Tn(x), Tn(y)) ≤ cnd(x, y) for all x and y, and that Then T has a unique fixed point. In applications, the existence and uniqueness of a fixed point often can be shown directly with the standard Banach fixed point theorem, by a suitable choice of the metric that makes the map T a contraction. Indeed, the above result by Bessaga strongly suggests to look for such a metric. See also the article on fixed point theorems in infinite-dimensional spaces for generalizations. A different class of generalizations arise from suitable generalizations of the notion of metric space, e.g. by weakening the defining axioms for the notion of metric. Some of these have applications, e.g., in the theory of programming semantics in theoretical computer science. See also Brouwer fixed-point theorem Caristi fixed-point theorem Contraction mapping Fichera's existence principle Fixed-point iteration Fixed-point theorems Infinite compositions of analytic functions Kantorovich theorem Notes References See chapter 7. Articles containing proofs Fixed-point theorems Metric geometry Topology