Metric Spaces

Abstract

The notion of the metric can be considered as a generalization of two point distance that was contrived systematically first by Euclid. In the modern mathematical set-up, Maurice René Frechét [116] is the first mathematician who axiomatically formulated the notion of metric space, under the name of L-space. On the other hand, first Felix Hausdorff [129] used the term “metric space” although he mainly focused on the role of point-sets within abstract set theory. Throughout the book, all sets are presumed nonempty. A distance function over a set X, namely, d: X× X→ [ 0, ∞), is called metric, or usual metric or standard metric if (d1) d(x, y) = d(y, x) = 0 ⟹ x= y ; (d2) d(x, x) = 0 ; (d3) d(x, y) = d(y, x) ; (d4) d(x, z) ≤ d(x, y) + d(y, z) ; for each x, y, z∈ X. In particular, the pair (X, d) is called metric space or usual metric space or standard metric space. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.