What is quasi metric space?
What is quasi metric space?
Introduction. A quasi-metric space is a set Z with a function ρ : Z ×Z → [0, ∞) which satisfies the conditions (1) ρ(z, z ) ≥ 0 for every z, z ∈ Z and ρ(z, z ) = 0 if and only if z = z ; (2) ρ(z, z ) = ρ(z ,z) for every z, z ∈ Z; (3) ρ(z, z ) ≤ K max{ρ(z, z ),ρ(z ,z )} for every z, z , z ∈ Z and some fixed K ≥ 1.
What is the difference between metric space and pseudo 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 T0 (i.e. distinct points are topologically distinguishable).
What is a discrete metric space?
metric space any set of points, the discrete metric specifies that the distance from a point to itself equal 0 while the distance between any two distinct points equal 1.
What is pseudo space?
Noun. pseudospace (countable and uncountable, plural pseudospaces) That which appears to be space or a space, or has only some aspects of space.
Is xy a metric space?
The answer is yes, and the theory is called the theory of metric spaces. A metric space is just a set X equipped with a function d of two variables which measures the distance between points: d(x, y) is the distance between two points x and y in X.
What is the difference between a metric and a quasi-metric?
The difference between a metric and a quasi-metric is that a quasi-metric does not possess the symmetry axiom (in the case we allow $d (x,y) e d (y,x)$ for some $x,y\\in \\mathbb X$ ). V. Schroeder, “Quasi-metric and metric spaces”.
Can Smyth complete quasi-metric spaces be characterized?
With the help of a suitable modification of its proof, we deduce a characterization of Smyth complete quasi-metric spaces which provides a quasi-metric generalization of the well-known characterization of metric completeness due to Kirk. Some illustrative examples are also given.
Is there a quasi-metric generalization of Caristi’s fixed point theorem?
We obtain a quasi-metric generalization of Caristi’s fixed point theorem for a kind of complete quasi-metric spaces.
