Definition of a Metric Space

begin quote: " 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". " end quote: See ita-3-1-1-page34.

begin quote: " 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, qE
(2)
d( p, q) = 0   if and only if   p = q
(3)
 d( p, q) = d(q, p)   for all   p, qE 
(4)
d( p, r)   ≤   d( p, q) + d( q, r)
for all p, q, rE
(triangle inequality)

" end quote: See ita-3-1-2-page34.

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begin quote: " Thus a metric space is an ordered pair  (E, d ),  where  E  is a set and  d  a function   d : E ⨯ ER   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, qE   we call   d( p, q )   the distance between   p   and   q;   d   itself is called the distance function or metric. " end quote: See ita-3-1-3-page34.

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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, qE,   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.)