Appendix to Chapter III
307
E is open ~ every x E E is interior to E,
F is closed ~ every x adherent to F belongs to F.
In consequence, E is open if and only if X - E is closed. Note that with
these definitions the set E = X is both open and closed, so similarly is the
empty set: in the latter case, all is satisfied since there is nothing to satisfy.
In a metric space, the open balls in the sense just defined are determined by
an inequality of the form d( a, x) < r, while the closed balls are defined by
d( a, x) ::; r, as one may verify using the triangle inequality.
The open or closed sets have the same properties as in R or C in respect
of unions and intersections, proved in the same way: the union of an arbitrary
family of open sets is an open set; the intersection of a finite family of open
sets is an open set, etc.
One should pay attention to the fact that the concept of open or closed
set is relative to a given "ambient" metric space X. If X is itself a subspace
of a larger metric space Y, a set E eX, to start with X itself, can very well
be open in X without being so in Y; there is no problem if X is open in Y.
In the general case, the open sets in X are (exercise!) the sets X n U, where
U is open in Y.
Even though these definitions have been inspired by familiar situations,
it is prudent not to let oneself be mystified by "obvious" geometrical images.
To give an example of what can happen, take for X the set Ql of rational
numbers, with the everyday distance d(x, y) = Ix - YI. For every a E Ql there
then exist balls in Ql with centre a which are simultaneously open and closed,
namely all the open balls B (a, r) where r is an irrational number. For B (a, r)
is the set of x E Ql (and not of x E R) such that Ix - al < r; every x E Ql
adherent to B(a,r) satisfies Ix - al ::; r, but since r is irrational one cannot
have Ix - al = r; so then Ix - al < r, whence x E B(a,r), qed.
In normal classical analysis one most often has to do with connected
spaces, i.e. in which the only simultaneously open and closed sets are the
whole set X itself and the empty set; more general are the locally connected
spaces, those in which every point has an open connected neighbourhood. It
is clear that every open subset of C is locally connected.
All the same there are very strange examples. If one considers the case of
a gyroscope whose axis of rotation is fixed at its lower extremity I, then its
upper end s, oscillating periodically, describes a curve C traced on a sphere
of centre I and contained between two horizontal sections of the sphere.
With great luck the trajectory of S will be a good closed curve (so compact)
with maybe some multiple points or cusps; for this it is necessary that after
completing an integral number of rotations about the vertical the gyroscope
should come back to the same position with the same velocity. But in the
general case the trajectory is not closed and it can well happen that the intersection of the curve C with every neighbourhood of every point of the sphere
lying between these horizontal limits is the union of a countable number of
307
E is open ~ every x E E is interior to E,
F is closed ~ every x adherent to F belongs to F.
In consequence, E is open if and only if X - E is closed. Note that with
these definitions the set E = X is both open and closed, so similarly is the
empty set: in the latter case, all is satisfied since there is nothing to satisfy.
In a metric space, the open balls in the sense just defined are determined by
an inequality of the form d( a, x) < r, while the closed balls are defined by
d( a, x) ::; r, as one may verify using the triangle inequality.
The open or closed sets have the same properties as in R or C in respect
of unions and intersections, proved in the same way: the union of an arbitrary
family of open sets is an open set; the intersection of a finite family of open
sets is an open set, etc.
One should pay attention to the fact that the concept of open or closed
set is relative to a given "ambient" metric space X. If X is itself a subspace
of a larger metric space Y, a set E eX, to start with X itself, can very well
be open in X without being so in Y; there is no problem if X is open in Y.
In the general case, the open sets in X are (exercise!) the sets X n U, where
U is open in Y.
Even though these definitions have been inspired by familiar situations,
it is prudent not to let oneself be mystified by "obvious" geometrical images.
To give an example of what can happen, take for X the set Ql of rational
numbers, with the everyday distance d(x, y) = Ix - YI. For every a E Ql there
then exist balls in Ql with centre a which are simultaneously open and closed,
namely all the open balls B (a, r) where r is an irrational number. For B (a, r)
is the set of x E Ql (and not of x E R) such that Ix - al < r; every x E Ql
adherent to B(a,r) satisfies Ix - al ::; r, but since r is irrational one cannot
have Ix - al = r; so then Ix - al < r, whence x E B(a,r), qed.
In normal classical analysis one most often has to do with connected
spaces, i.e. in which the only simultaneously open and closed sets are the
whole set X itself and the empty set; more general are the locally connected
spaces, those in which every point has an open connected neighbourhood. It
is clear that every open subset of C is locally connected.
All the same there are very strange examples. If one considers the case of
a gyroscope whose axis of rotation is fixed at its lower extremity I, then its
upper end s, oscillating periodically, describes a curve C traced on a sphere
of centre I and contained between two horizontal sections of the sphere.
With great luck the trajectory of S will be a good closed curve (so compact)
with maybe some multiple points or cusps; for this it is necessary that after
completing an integral number of rotations about the vertical the gyroscope
should come back to the same position with the same velocity. But in the
general case the trajectory is not closed and it can well happen that the intersection of the curve C with every neighbourhood of every point of the sphere
lying between these horizontal limits is the union of a countable number of