310
Appendix to Chapter III
Cantor and Meray (Chap. III, end of nO 10). To do this, one considers the
set S of all Cauchy sequences in X - it is contained in the set of maps of N
into X - and one defines an equivalence relation R on S by declaring that
two Cauchy sequences (xn) and (Yn) are equivalent if limd (xn, Yn) = 0; see
the end of Chap. I, nO 4, for the general concept of equivalence, specifically
invented for the kind of situation examined here; the conditions of the general
definition are satisfied here by reason of the triangle inequality
Now to every Cauchy sequence we associate its equivalence class, the subset
of S formed by all the sequences equivalent to the given sequence. The set of
these classes, Le. the quotient set S / R, is by definition X. The fact that an
element of X can be a very complicated set is nothing abnormal and, quite
to the contrary, confirms the basic postulate of Chap. I: every mathematical
object is a set, even when this is not immediately obvious, so a fortiori in
the present case.
If two Cauchy sequences (xn) and (Yn) in X define two elements x and Y
of X one puts
(3.5)
Since d (xp, Xq) and d (YP' Yq) are < r for p and q large, the relation
shows that the sequence of d (xn, Yn) satisfies Cauchy's criterion, whence the
existence of the limit (5); similar inequalities show that it is independent of
the Cauchy sequences chosen in X to "represent" x and y. The fact that
d(x, y) = 0 ¢:::::} x = Y does not have to be proved: it is the very definition
of equality between classes of Cauchy sequences; the other properties of distances are obvious. The "embedding" of X into X is obtained by associating
to each x E X the class ofthe Cauchy sequence (x, x, X, •• • ), Le. the set of sequences (xn) which converge to x in X. In practice, one makes no distinction
between an element of X and its class in X (which confirms the fact that,
once a mathematical object is defined as too complicated a set, one stops
thinking of this psychologically inhibiting "detail" ... ).
It is also clear that, for x, Y EX, the distance between x and Y as
calculated in X, for example starting from the sequences (x, x, x, ... ) and
(y, y, y, .. . ), is the same as in X.
To calculate the distance between an x E X and ayE X one chooses
a Cauchy sequence (xn) in the class x; since Y corresponds to the sequence
with general term y, the definition (5) shows that d(x,y) = limd(xn,Y); in
particular, one has d (x, x p) = limn d (xm xp), a result < r for p large since
(xn) is a Cauchy sequence; the distance from x to the points xp E X thus
Appendix to Chapter III
Cantor and Meray (Chap. III, end of nO 10). To do this, one considers the
set S of all Cauchy sequences in X - it is contained in the set of maps of N
into X - and one defines an equivalence relation R on S by declaring that
two Cauchy sequences (xn) and (Yn) are equivalent if limd (xn, Yn) = 0; see
the end of Chap. I, nO 4, for the general concept of equivalence, specifically
invented for the kind of situation examined here; the conditions of the general
definition are satisfied here by reason of the triangle inequality
Now to every Cauchy sequence we associate its equivalence class, the subset
of S formed by all the sequences equivalent to the given sequence. The set of
these classes, Le. the quotient set S / R, is by definition X. The fact that an
element of X can be a very complicated set is nothing abnormal and, quite
to the contrary, confirms the basic postulate of Chap. I: every mathematical
object is a set, even when this is not immediately obvious, so a fortiori in
the present case.
If two Cauchy sequences (xn) and (Yn) in X define two elements x and Y
of X one puts
(3.5)
Since d (xp, Xq) and d (YP' Yq) are < r for p and q large, the relation
shows that the sequence of d (xn, Yn) satisfies Cauchy's criterion, whence the
existence of the limit (5); similar inequalities show that it is independent of
the Cauchy sequences chosen in X to "represent" x and y. The fact that
d(x, y) = 0 ¢:::::} x = Y does not have to be proved: it is the very definition
of equality between classes of Cauchy sequences; the other properties of distances are obvious. The "embedding" of X into X is obtained by associating
to each x E X the class ofthe Cauchy sequence (x, x, X, •• • ), Le. the set of sequences (xn) which converge to x in X. In practice, one makes no distinction
between an element of X and its class in X (which confirms the fact that,
once a mathematical object is defined as too complicated a set, one stops
thinking of this psychologically inhibiting "detail" ... ).
It is also clear that, for x, Y EX, the distance between x and Y as
calculated in X, for example starting from the sequences (x, x, x, ... ) and
(y, y, y, .. . ), is the same as in X.
To calculate the distance between an x E X and ayE X one chooses
a Cauchy sequence (xn) in the class x; since Y corresponds to the sequence
with general term y, the definition (5) shows that d(x,y) = limd(xn,Y); in
particular, one has d (x, x p) = limn d (xm xp), a result < r for p large since
(xn) is a Cauchy sequence; the distance from x to the points xp E X thus