306
Appendix to Chapter III
2 - Open and closed sets
In a metric space, as in JR or C, one can define an open ball B (a, r) with
centre a by the strict inequality d( a, x) < r, where r is a number > o. The
weak inequality d( a, x) ::::; r, with r ~ 0, defines a closed ball, including for
r = 0 if one maintains that every intersection of closed balls with centre a is
again a closed ball. Although, in JR, the intersection of two open balls with
arbitrary centres a' and a" is again an open ball, this is not so in the general
case, nor even in JR2; the triangle inequality, however, shows that if a E
B(a', r') n B(a", r") there exists a ball B(a, r) contained in this intersection.
The analogous statement for closed balls is false, even in JR. The definitions of
Chap. II, nO 3 in respect of what happens on a neighbourhood of a point a or
at infinity extends immediately to metric spaces. An assertion P(x) involving
a variable x E X is true on a neighbourhood of an a E X if there exists an open
ball B with centre a such that x E B ==? P(x). It is true at infinity if, having
chosen acE X, there exists a number R such that d( c, x) > R ==? P( x);
the choice of c hardly matters. Since the intersection of a finite number of
open balls with centre a is again an open ball with centre a, it is clear that
if fifteen assertions are separately valid on a neighbourhood of a, they are
simultaneously valid on a neighbourhood of a.
A yet more convenient way of expressing this is to introduce, with N. Bourbaki, the technical, not naive, concept, of a neighbourhood of a point a in a
metric space X. By definition, it is any set containing an open ball with
centre a; it may reduce to this ball or spill over ad libitum. It is clear that
(i) any set containing a neighbourhood of a is again a neighbourhood of a,
(ii) the intersection of a finite number of neighbourhoods of a is a neighbourhood of a,
(iii) any neighbourhood of an a E X is a neighbourhood of all b E X sufficiently close to a,
because an open ball B(a) with centre a contains an open ball with centre b
for any bE B(a), so is a neighbourhood of b.
Now, the most radical way of expressing that an assertion P(x), where
x varies in X, is "true for all x sufficiently close to a" is to say: the set of
x E X such that P(x) is true is a neighbourhood of a. In this form, the r,
the c, the 8, etc. have disappeared.
As in JR or C, the points a of X may, relative to a set E eX, be divided
into three categories: the interior points of E (E contains an open ball with
centre a), the exterior points of E (they are interior to X - E) and the
boundary points (every open ball with centre a meets E and X - E). The
points which are not exterior to E are called adherent to E; one writes E for
the set of adherent points the closure of E.
These definitions lead immediately, as in the case of R or C, to the concepts of open set and of closed set in a metric space X:
Appendix to Chapter III
2 - Open and closed sets
In a metric space, as in JR or C, one can define an open ball B (a, r) with
centre a by the strict inequality d( a, x) < r, where r is a number > o. The
weak inequality d( a, x) ::::; r, with r ~ 0, defines a closed ball, including for
r = 0 if one maintains that every intersection of closed balls with centre a is
again a closed ball. Although, in JR, the intersection of two open balls with
arbitrary centres a' and a" is again an open ball, this is not so in the general
case, nor even in JR2; the triangle inequality, however, shows that if a E
B(a', r') n B(a", r") there exists a ball B(a, r) contained in this intersection.
The analogous statement for closed balls is false, even in JR. The definitions of
Chap. II, nO 3 in respect of what happens on a neighbourhood of a point a or
at infinity extends immediately to metric spaces. An assertion P(x) involving
a variable x E X is true on a neighbourhood of an a E X if there exists an open
ball B with centre a such that x E B ==? P(x). It is true at infinity if, having
chosen acE X, there exists a number R such that d( c, x) > R ==? P( x);
the choice of c hardly matters. Since the intersection of a finite number of
open balls with centre a is again an open ball with centre a, it is clear that
if fifteen assertions are separately valid on a neighbourhood of a, they are
simultaneously valid on a neighbourhood of a.
A yet more convenient way of expressing this is to introduce, with N. Bourbaki, the technical, not naive, concept, of a neighbourhood of a point a in a
metric space X. By definition, it is any set containing an open ball with
centre a; it may reduce to this ball or spill over ad libitum. It is clear that
(i) any set containing a neighbourhood of a is again a neighbourhood of a,
(ii) the intersection of a finite number of neighbourhoods of a is a neighbourhood of a,
(iii) any neighbourhood of an a E X is a neighbourhood of all b E X sufficiently close to a,
because an open ball B(a) with centre a contains an open ball with centre b
for any bE B(a), so is a neighbourhood of b.
Now, the most radical way of expressing that an assertion P(x), where
x varies in X, is "true for all x sufficiently close to a" is to say: the set of
x E X such that P(x) is true is a neighbourhood of a. In this form, the r,
the c, the 8, etc. have disappeared.
As in JR or C, the points a of X may, relative to a set E eX, be divided
into three categories: the interior points of E (E contains an open ball with
centre a), the exterior points of E (they are interior to X - E) and the
boundary points (every open ball with centre a meets E and X - E). The
points which are not exterior to E are called adherent to E; one writes E for
the set of adherent points the closure of E.
These definitions lead immediately, as in the case of R or C, to the concepts of open set and of closed set in a metric space X: