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If M is non-empty compact set, then the metric projection p M is upper semi-continuous, but might not be lower semi-continuous. But if X is a normed space and M is a finite-dimensional Chebyshev set, then p M is continuous. [citation needed] Moreover, if X is a Hilbert space and M is closed and convex, then p M is Lipschitz continuous with ...
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. [1] Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.
A set with a topology is called a topological space. Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.
Let X and Y be two metric spaces, and F a family of functions from X to Y.We shall denote by d the respective metrics of these spaces.. The family F is equicontinuous at a point x 0 ∈ X if for every ε > 0, there exists a δ > 0 such that d(ƒ(x 0), ƒ(x)) < ε for all ƒ ∈ F and all x such that d(x 0, x) < δ.
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.That is, a topological space (,) is said to be metrizable if there is a metric: [,) such that the topology induced by is . [1] [2] Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.
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.
Let (,) be a metric space, where is a given set. For any point and any non-empty subset , define the distance between the point and the subset: (,):= (,),.For any sequence of subsets {} = of , the Kuratowski limit inferior (or lower closed limit) of as ; is := {:,} = {: (,) =}; the Kuratowski limit superior (or upper closed limit) of as ; is := {:,} = {: (,) =}; If the Kuratowski limits ...
Although continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of neighbourhoods of distinct points, so it requires a metric space, or more generally a uniform space.