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The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied for k ≤ 1, then the mapping is said to be a non-expansive map. More generally, the idea of a contractive mapping can be defined for maps between metric spaces.
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to find those fixed points.
A variant of the closed map lemma states that if a continuous function between locally compact Hausdorff spaces is proper then it is also closed. In complex analysis, the identically named open mapping theorem states that every non-constant holomorphic function defined on a connected open subset of the complex plane is an open map.
A degree two map of a sphere onto itself.. In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping.
In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine's definition , is such a function that maps convergent sequences into convergent sequences: if x n → x then g ( x n ) → g ( x ).
For instance, any non-empty space retracts to a point in the obvious way (any constant map yields a retraction). If X is Hausdorff , then A must be a closed subset of X . If r : X → A {\textstyle r:X\to A} is a retraction, then the composition ι∘ r is an idempotent continuous map from X to X .
For 0 < i < n, any mapping from S i to S n is homotopic (i.e., continuously deformable) to a constant mapping, i.e., a mapping that maps all of S i to a single point of S n. In the smooth case, it follows directly from Sard's Theorem. Therefore the homotopy group is the trivial group.
In complex analysis, the open mapping theorem states that if is a domain of the complex plane and : is a non-constant holomorphic function, then is an open map (i.e. it sends open subsets of to open subsets of , and we have invariance of domain.).