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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.
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 contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some real number < such that for all x and y in M,
The product of a finite set of metric spaces in Met is a metric space that has the cartesian product of the spaces as its points; the distance in the product space is given by the supremum of the distances in the base spaces. That is, it is the product metric with the sup norm. However, the product of an infinite set of metric spaces may not ...
A metric space M is compact if every open cover has a finite subcover (the usual topological definition). A metric space M is compact if every sequence has a convergent subsequence. (For general topological spaces this is called sequential compactness and is not equivalent to compactness.) A metric space M is compact if it is complete and ...
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.
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.
In particular, if X is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in ...