THE RISE OF FUNCTIONS
17
(ɪɜ) {D2,f2} ≡ {C1(h>),∕2}.
Now there is no conformal mapping from the disc (11) into all of C1, and
thus the two ensembles (10), as given by (12) and (13), cannot be homeo-
morphic.
The conclusion to be drawn from this counterexample is this: that the
Weierstrass continuation process does not at all apply to an “abstract”
ensemble
(14) {D,f},
in which D is an “abstract” Riemann surface and f a holomorphic function
on it. But if one does start out with an abstract ensemble (14), then the
process does become applicable if there is given an additional Riemann
surface S (which in the “classical” case of Weierstrass is the complex plane
or the Gaussian sphere, but which, in fact, can be quite general) and a
holomorphic Iinbranched mapping of D into S. In fact, if we denote this
mapping by g, then the process applies not to the ensemble (14) but to the
greater ensemble
(15) {D,f,g),
and the assertion is that this greater ensemble has a continuation
(16) {D,f,g}
which is both maximal and unique. Also, the symbol /in (15) need not
represent a single holomorphic function, but may represent an assemblage
of holomorphic functions, or, in fact, of scalar or tensorial holomorphic
objects, and may also subsume mappings into some complex spaces, which
need not have anything to do with the fixed space S'. Also, the scalar or
tensorial objects need not be strictly holomorphic, but they may also be
meromorphic, provided they are so both in (15) and (16).
All this follows readily by adapting the reasoning in Chapter 1 of Weyl’s
book [14]. It also applies to the case of several complex variables, if D, S
are equidimensional and they and their extensions are assumed to be arcwise
connected. It should be noted that for n ≥ 2 general complex spaces like
D, S need not be separable, and if they are indeed not separable then
uncountably many sheets of D may be lying over the same points of S.
If however D and 5 happen to be separable, then D will be so too, and
there will be only countably many sheets of D spread over 5.
We have noted before that the object f in (15) may subsume general