Some Notes About the Cardinals of Metric Space

Theorem 1 Let X be a nonempty countable set, K={: is a discrete metric space}, define ≌ iff((?)f) (f is an equilong isomorphism from to , for a given ∈K, define = { ∈K: ≌}. Let C={: ∈K},then |C|=|K|=|{d:d is a metric on X}|=2~((?)0) The Theorem 2 illustrates that there exists a nonempty countable set X on which we can define 2~((?)0) nondiscrete metric spaces such that they are not isomorphic each other.