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:
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M1: d(x,x) = 0
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M2: d(x,z ≤ d(x,y) + d(y,z)
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M3: d(x,y) = d(y,x)
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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) < e}. A topological space together with a metric giving its topology is called a metric space. Although a single metric wall yield a unique topology on a given set, it is possible to find more than one metric which will yield the same topology. In fact, there are always an infinite number of metrics which will yield the same metric space (Example 134).
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© 1978 Springer-Verlag New York Inc.
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Steen, L.A., Seebach, J.A. (1978). Metric Spaces. In: Counterexamples in Topology. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-6290-9_5
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DOI: https://doi.org/10.1007/978-1-4612-6290-9_5
Publisher Name: Springer, New York, NY
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