54
VIII – Cauchy Theory
that for any compact set K ⊂ C, z n ∈ C − K for large n. With respect to this
topology, ˆ
C is compact. If indeed ˆ
C is the union of a family of open subsets
U i , one of them, say U j , contains the point at infinity, and hence also the
exterior of a compact set K ⊂ C; as K can be covered (BL) by finitely may
sets C ∩ U i , these U i , together with U j , form a finite cover of ˆ
C, qed. BW can
also be checked.
To understand the topology of ˆ
C “ geometrically ”, consider the classical
unit sphere S
2 in R
3 = C × R and, denoting its north pole by ν = (0, 0, 1),
consider the map p associating to any ζ ∈ S
2 other than ν the point z ∈ C =
R
2 , where the line passing through ν and ζ meets the equatorial plane C; this
is the “ stereographic projection ” from the north pole used to map regions
not too close to it. We thus get a homeomorphism from S
2
− {ν} onto C
transforming the exterior of a disc of radius R centered at 0 in C into a set
of points ζ ∈ S
2
− {ν} whose third coordinate satisfies a relation a < ζ 3 < 1.
Therefore, if we generalize the definition of p by setting p(ν) = ∞, we obtain
a continuous bijection from S
2 onto ˆ
C, hence a homeomorphism since the
sphere is compact. ˆ
C is generally called the la Riemann sphere; I do not know
whether he would have appreciated this tribute : it is like congratulating an
Olympic cycling champion for having won the amateur criterion of his home
town.
ˆ
C is indeed the only trivial example of a compact “ Riemann surface ” or,
in today’s terminology, of a “ compact complex analytic manifold of dimension 1 ” (Chap. X) : A reasonable definition of the notion of a holomorphic
function on an open subset U of S
2 follows by stipulating that such a function should depend holomorphically, possibly also at infinity, on the point
p(ζ) ∈ p(U ), where p : S
2
−→ ˆ
C is the stereographic projection. Without
resorting to a cartography unlikely to yield useful generalizations in this type
of context,
35 a function f defined on an open subset U of ˆ
C with values in
C is said to be holomorphic if it is so in the usual sense when U ⊂ C, and in
case ∞ ∈ U , if it is so in the usual sense on U ∩ C and approaches a finite
limit f (∞) at infinity; as the value f (∞) is defined by (17), this amounts to
saying that f (z) = g(1/z), where g is holomorphic in the neighbourhood of 0.
Besides, classical definitions together with those given in (iv) for behaviour
at ∞ makes it possible to give meaning to the notion of a “ pole ” of a function when it is holomorphic in the neighbourhood of a point a ∈ ˆ
C, except
35 The construction of ˆ
C can be generalized to any locally compact space X : the
open subsets of ˆ
X = X ∪ {∞} containing the point ∞ are set to be the complements of the compact subsets of X. Thus X becomes the complement of a point
in the compact space ˆ
X, the Alexandrov one-point compactification of X. For
X = R, the space obtained is homeomorphic to the unit circle T. This can be
seen by using the map t → (t − i)/(t + i) from R to T − {1}; it can be extended
by continuity to ˆ
R if a value of 1 is set for t = ∞, and as it is then bijective
and continuous, it is necessarily a homeomorphism. This construction transforms
functions approaching a limit at infinity into functions on ˆ
X continuous at ∞.
This is low level, but sometimes useful, general topology.
VIII – Cauchy Theory
that for any compact set K ⊂ C, z n ∈ C − K for large n. With respect to this
topology, ˆ
C is compact. If indeed ˆ
C is the union of a family of open subsets
U i , one of them, say U j , contains the point at infinity, and hence also the
exterior of a compact set K ⊂ C; as K can be covered (BL) by finitely may
sets C ∩ U i , these U i , together with U j , form a finite cover of ˆ
C, qed. BW can
also be checked.
To understand the topology of ˆ
C “ geometrically ”, consider the classical
unit sphere S
2 in R
3 = C × R and, denoting its north pole by ν = (0, 0, 1),
consider the map p associating to any ζ ∈ S
2 other than ν the point z ∈ C =
R
2 , where the line passing through ν and ζ meets the equatorial plane C; this
is the “ stereographic projection ” from the north pole used to map regions
not too close to it. We thus get a homeomorphism from S
2
− {ν} onto C
transforming the exterior of a disc of radius R centered at 0 in C into a set
of points ζ ∈ S
2
− {ν} whose third coordinate satisfies a relation a < ζ 3 < 1.
Therefore, if we generalize the definition of p by setting p(ν) = ∞, we obtain
a continuous bijection from S
2 onto ˆ
C, hence a homeomorphism since the
sphere is compact. ˆ
C is generally called the la Riemann sphere; I do not know
whether he would have appreciated this tribute : it is like congratulating an
Olympic cycling champion for having won the amateur criterion of his home
town.
ˆ
C is indeed the only trivial example of a compact “ Riemann surface ” or,
in today’s terminology, of a “ compact complex analytic manifold of dimension 1 ” (Chap. X) : A reasonable definition of the notion of a holomorphic
function on an open subset U of S
2 follows by stipulating that such a function should depend holomorphically, possibly also at infinity, on the point
p(ζ) ∈ p(U ), where p : S
2
−→ ˆ
C is the stereographic projection. Without
resorting to a cartography unlikely to yield useful generalizations in this type
of context,
35 a function f defined on an open subset U of ˆ
C with values in
C is said to be holomorphic if it is so in the usual sense when U ⊂ C, and in
case ∞ ∈ U , if it is so in the usual sense on U ∩ C and approaches a finite
limit f (∞) at infinity; as the value f (∞) is defined by (17), this amounts to
saying that f (z) = g(1/z), where g is holomorphic in the neighbourhood of 0.
Besides, classical definitions together with those given in (iv) for behaviour
at ∞ makes it possible to give meaning to the notion of a “ pole ” of a function when it is holomorphic in the neighbourhood of a point a ∈ ˆ
C, except
35 The construction of ˆ
C can be generalized to any locally compact space X : the
open subsets of ˆ
X = X ∪ {∞} containing the point ∞ are set to be the complements of the compact subsets of X. Thus X becomes the complement of a point
in the compact space ˆ
X, the Alexandrov one-point compactification of X. For
X = R, the space obtained is homeomorphic to the unit circle T. This can be
seen by using the map t → (t − i)/(t + i) from R to T − {1}; it can be extended
by continuity to ˆ
R if a value of 1 is set for t = ∞, and as it is then bijective
and continuous, it is necessarily a homeomorphism. This construction transforms
functions approaching a limit at infinity into functions on ˆ
X continuous at ∞.
This is low level, but sometimes useful, general topology.