Definition of a Metric Space
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A metric space is a set together with a rule which associates with each pair of elements of the set a real number such that certain axioms are satisfied. The axioms are chosen in such a manner that it is reasonable to think of the set as a "space", and the real number associated with two elements of the set as the "distance between two points".
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See ita311page34.
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Definition. A metric space is a set E, together with a rule which associates with each pair p, q ∈ E a real number d (p, q) such that
Number 
Condition 
(1) 
d ( p, q) ≥ 0 for all p, q ∈ E

(2) 
d ( p, q) = 0 if and only if p = q

(3) 
d ( p, q) = d ( q, p) for all p, q ∈ E

(4) 
d ( p, r) ≤ d ( p, q) + d ( q, r)
for all p, q, r ∈ E
(triangle inequality)

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See ita312page34.
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Thus a metric space is an ordered pair (E, d ), where E is a set and d a function d : E ⨯ E → R satisfying properties (1)  (4). In dealing with a metric space (E, d ) it is often understood from the context what d is, or that a certain specific d is to be borne in mind, and then one often speaks simply of "the metric space E "; this is logically incorrect but very convenient. The elements p, q, r, ... of a metric space E (to be absolutely correct we should say "the elements of the underlying set E of the metric space (E, d )", but let us not be too pedantic) are called the points of E, and if p, q ∈ E we call d( p, q ) the distance between p and q; d itself is called the distance function or metric.
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See ita313page34.
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Remark. It seems desirable to include the null set as a possible "set of points" in a metric space. We can do this if we regard the conditions (1) to (4) above as material implications. The first three conditions would then begin with "If p, q ∈ E, then ...", and similarly with the fourth condition. Then we have E = ∅ and the distance function reduces to the null set. (See the analysis of a function from the cross product of the null set with itself into the null set which was presented in connection with the real number system.) Then our metric space becomes (∅, ∅), but this is valid because the conditions (1) to (4) have been stated as material implications: No matter what p and q are, any conditional statement that begins with "If p, q ∈ ∅, then..." must be true, since the premise is false. (See the truth table.)