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Algebras of Holomorphic Functions in Ringed Spaces, I
Published online by Cambridge University Press: 20 November 2018
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A pair (
) is a ringed space if it is a subsheaf of rings with 1 of the sheaf of germs of continuous functions on X. If U is an open subset of X, we denote the set of sections over U relative to 
by 
. If 
, then 
implies that there exists some open neighbourhood V of u, V ⊂ U, and some g continuous on V such that the germ of g at u, ug is ϕ(u). Now we define ϕ(u) (u) to be g(u) and in this way we obtain, in a unique fashion, a continuous complex-valued function on U. The collection of all such functions for a given set is denoted by 
and is called the -holomorphic functions on U.
THEOREM. Let X be a locally connected Hausdorff space and () a ringed space.
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- Copyright © Canadian Mathematical Society 1969
References
1.
Bochner, S. and Martin, W., Several complex variables, Princeton Mathematical Series, Vol. 10 (Princeton Univ. Press, Princeton, N.J., 1948).Google Scholar
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Hoffman, K., Domains of holomorphy, Mimeographed notes, Massachusetts Institute of Technology, Cambridge, Mass., 1958.Google Scholar
5.
Quigley, F., Lectures on several complex variables, Tulane University, New Orleans, Louisiana, 1964-65, 1965-66.Google Scholar
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