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Functions to a metric space. If X is any set and M is a metric space, then the set of all bounded functions: (i.e. those functions whose image is a bounded subset of ) can be turned into a metric space by defining the distance between two bounded functions f and g to be (,) = ((), ()).
Cartan connection. Cartan-Hadamard space is a complete, simply-connected, non-positively curved Riemannian manifold.. Cartan–Hadamard theorem is the statement that a connected, simply connected complete Riemannian manifold with non-positive sectional curvature is diffeomorphic to R n via the exponential map; for metric spaces, the statement that a connected, simply connected complete ...
Thus metric spaces together with metric maps form a category Met. Met is a subcategory of the category of metric spaces and Lipschitz functions. A map between metric spaces is an isometry if and only if it is a bijective metric map whose inverse is also a metric map. Thus the isomorphisms in Met are precisely the isometries.
The function d is a metric on M, and d(x, y) is the distance between x and y. The collection of all open balls of M is a base for a topology on M; this is the topology on M induced by d. Every metric space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metric space is first-countable. Metrizable/Metrisable
For any metric space M, it is possible to construct a complete metric space M′ (which is also denoted as ¯), which contains M as a dense subspace. It has the following universal property : if N is any complete metric space and f is any uniformly continuous function from M to N , then there exists a unique uniformly continuous function f ...
The Assouad–Nagata dimension of a metric space (X, d) is defined as the smallest integer n for which there exists a constant C > 0 such that for all r > 0 the space X has a Cr-bounded covering with r-multiplicity at most n + 1.
Let (M, d) be a metric space, let K be a subset of M, and let r be a positive real number.Let B r (x) denote the ball of radius r centered at x.A subset C of M is an r-external covering of K if:
The Assouad dimension of a metric space is always greater than or equal to its upper box dimension, which in turn is greater than or equal to the Hausdorff dimension. [5] The Lebesgue covering dimension of a metrizable space X is the minimal Assouad dimension of any metric on X. In particular, for every metrizable space there is a metric for ...