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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 ...
The space C [a, b] of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm. However, the supremum norm does not give a norm on the space C ( a , b ) of continuous functions on ( a , b ) , for it may contain unbounded functions .
A metric space (,) is said to be uniformly discrete if there exists a packing radius > such that, for any ,, one has either = or (,) >. [1] The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set {:}.
A compact metric space (X, d) also satisfies the following properties: Lebesgue's number lemma: For every open cover of X, there exists a number δ > 0 such that every subset of X of diameter < δ is contained in some member of the cover. (X, d) is second-countable, separable and Lindelöf – these three conditions are equivalent for metric ...
Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space.
The difference between pseudometrics and metrics is entirely topological. That is, a pseudometric is a metric if and only if the topology it generates is T 0 (that is, distinct points are topologically distinguishable). The definitions of Cauchy sequences and metric completion for metric spaces carry over to pseudometric spaces unchanged. [5]
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,
In mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning. It is also a useful step toward the construction of the metric envelope, or tight span, which are basic (injective) objects of the category of metric spaces.