What is B-metric space?
What is B-metric space?
Definition 1 Let X be a set and let d be a function from X × X into [0, ∞). Then (X, d) is said to be a b-metric space if the following hold: d(x, y) = 0 ⇔ x = y; d(x, y) = d(y, x) (symmetry); There exists K ≥ 1 satisfying d(x, z) ≤ K(d(x, y) + d(y, z)) for any x, y, z ∈ X (K-relaxed triangle inequality).
Who introduced B-metric space?
1. Introduction. The idea of b-metric was initiated from the works of Bourbaki [1] and Bakhtin [2]. Czerwik [3] gave an axiom which was weaker than the triangular inequality and formally defined a b-metric space with a view of generalizing the Banach contraction mapping theorem.
What is metric space with example?
A metric space is a set X together with such a metric. The prototype: The set of real numbers R with the metric d(x, y) = |x – y|. This is what is called the usual metric on R. The complex numbers C with the metric d(z, w) = |z – w|.
What are the types of metric space?
Contents
- 5.1 Complete spaces.
- 5.2 Bounded and totally bounded spaces.
- 5.3 Compact spaces.
- 5.4 Locally compact and proper spaces.
- 5.5 Connectedness.
- 5.6 Separable spaces.
- 5.7 Pointed metric spaces.
How do you show metric space?
Show that the real line is a metric space. Solution: For any x, y ∈ X = R, the function d(x, y) = |x − y| defines a metric on X = R. It can be easily verified that the absolute value function satisfies the axioms of a metric.
How do you show a metric space?
To verify that (S, d) is a metric space, we should first check that if d(x, y) = 0 then x = y. This follows from the fact that, if γ is a path from x to y, then L(γ) ≥ |x − y|, where |x − y| is the usual distance in R3. This implies that d(x, y) ≥ |x − y|, so if d(x, y) = 0 then |x − y| = 0, so x = y.
What is a metric space?
metric space, in mathematics, especially topology, an abstract set with a distance function, called a metric, that specifies a nonnegative distance between any two of its points in such a way that the following properties hold: (1) the distance from the first point to the second equals zero if and only if the points …
Which is not a metric space?
Technically a metric space is not a topological space, and a topological space is not a metric space: a metric space is an ordered pair ⟨X,d⟩ such that d is a metric on X, and a topological space is an ordered pair ⟨X,τ⟩ such that τ is a topology on X.
Why do we need metric space?
In mathematics, a metric space is a set where a distance (called a metric) is defined between elements of the set. Metric space methods have been employed for decades in various applications, for example in internet search engines, image classification, or protein classification.
Are metric spaces hard?
Metric spaces are more general than normed spaces, because they need not be vector spaces. They are easier than general topological spaces, but introduce all of the relevant concepts.