Metric Spaces

Lynn Arthur Steen and J. Arthur Seebach
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Lynn Arthur Steen: Saint Olaf College
J. Arthur Seebach: Saint Olaf College

Chapter Section 5 in Counterexamples in Topology, 1978, pp 34-38 from Springer

Abstract: Abstract A metric for a set X is a mapping d of X × X into the nonnegative real numbers satisfying the following conditions for all x,y,z ∈ X: M1: d(x,x) = 0 M2: d(x,z ≤ d(x,y) + d(y,z) M3: d(x,y) = d(y,x) M4: if x ≠ y, d(x,y) > 0. We call d(x,y) the distance between x and y. If d satisfies only M1, M2, and M4 it is called quasimetric, while if it satisfies M1, M2, and M3 it is called a pseudometric. It is possible to use a metric to define a topology on X by taking as a basis all open balls B(x,e) = {y ∈ X|d(x,y)

Date: 1978
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DOI: 10.1007/978-1-4612-6290-9_5

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