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Pertinent Properties of Euclidean Space
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
A consequence of example 5 is that every metric space is also a topological space with the topology induced by the metric. The converse is, in general, not true. There exist topological spaces X with topology τ that are not metrizable; i.e., it is impossible to define a metric d on the set X such d induces τ. Furthermore, in view of these examples it is easy to see that a given topology may have many different bases that will generate it. This is analogous to the concept of a basis for a vector space: Different bases can generate the same vector space. Any linearly independent set of n vectors in ℝn can be used as a basis for the vector space ℝn.
b-Metric Spaces
Published in Dhananjay Gopal, Praveen Agarwal, Poom Kumam, Metric Structures and Fixed Point Theory, 2021
Nguyen Van Dung, Wutiphol Sintunavarat
It follows from Example 1, and some others, that there exists a b-metric which is not a metric. However, the class of all b-metric spaces and the class of all metric spaces are coincident in the sense that every b-metric space is a metrizable space. Recall that the metrization problem is concerned with conditions under which a topological space X is metrizable [23], where for a function D : X × X → [0,∞) satisfying some axioms and generating a topology T on X, and for a metric D : X × X → [0,∞), the topological space (X,T ) is called metrizable by the metric D if T and the metric topology induced by D coincide. Recall that a space X is a metric space if there exists a metric D : X × X → [0,∞) that satisfies the following conditions for all x,y,z ∈ X. D(x,y) = 0 if and only if x = y.The symmetry: D(x,y) = D(y,x).The triangle inequality: D(x,z) ≤ D(x,y) + D(y,z).Some generalizations of the triangle inequality (III) were introduced such asThe generalized triangle inequality: If D(x,y) < ε and D(y,z) < ε then D(x,z) < 2ε.The uniform regular property: For every ε > 0 there exists ϕ(ε) > 0 such that if D(x,y) < ϕ(ε) and D(y,z) < ϕ(ε) then D(x,z) ≤ ε.
A random weak ergodic property of infinite products of operators in metric spaces
Published in Optimization, 2019
Simeon Reich, Alexander J. Zaslavski
Denote by the set of all sequences . The set is equipped with the uniformity determined by the base where . It is not difficult to see that the uniform space is metrizable (by a metric ) and complete. The metric induces a topology in . Let be a closed subset of and let it also be equipped with the metric . This metric induces the topology in . We assume that for each and each mapping , we have We say that the sequence has the random weak ergodic property (RWEP, for short) if for each , there exists a neighbourhood of in and a natural number n such that for each , each integer , each and each , We are now ready to formulate our main result.