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Some Notes About the Cardinals of Metric Space
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[1] Teaching and Research Section of Math
[2] Hangzhou Institute of Electronical Engineering
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Some Notes About the Cardinals of Metric Space;
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Theorem 1 Let X be a nonempty countable set, K={<X,d>: <X, d> is a discrete metric space}, define <X, d>≌<X, d’> iff(（?）f) (f is an equilong isomorphism from <X,a> to <x,a’>, for a given <x, d> ∈K, define<x, d> = {<x,d’> ∈K: <X,d’>≌<x,d>}. Let C={<x,d>: <X,d> ∈K},then |C|=|K|=|{d:d is a metric on X}|=2The Theorem 2 illustrates that there exists a nonempty countable set X on which we can define 2nondiscrete metric spaces such that they are not isomorphic each other.
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