The theory of Riemann surfaces has a geometric and an analytic part. The former deals with the axiomatic definition of a Riemann surface, methods of. Riemann Surfaces. Front Cover. Lars V. Ahlfors, Leo Sario. Princeton University Press, Jan 1, – Mathematics – pages. A detailed exposition, and proofs, can be found in Ahlfors-Sario , Forster Riemann Surface Meromorphic Function Elliptic Curve Complex Manifold.
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Lars V. Ahlfors, L. Sario-Riemann Surfaces
The requirements are interpreted to hold also dario the empty collection. In the case of a aubaet it is convenient to replace the relatively open seta of a covering by corresponding open seta of the whole apace. We have already pointed out that the components are closed, independently of the local connectedness.
Sometimes it is more convenient to express compactneea in terms of closed seta. Surfacse, Damasceno, Awada – Pronto-socorro Pronto-socorro: From Q c 01 it follows that Oa is empty, and hence Q is connected. At the very end of the chapter it is then shown, by essential use of the Jordan curve theorem, that every surface which satisfies the second axiom of countability permits a.
The complement of an open set is said to be cloftd. The following theorem is thus merely a rephrasing of the definition. Suppose that p belongs to the component 0, and let V p be a connected neighborhood. Surface acetylation of bacterial cellulose Surface acetylation of bacterial cellulose. A topology 9″ riemabn is said to be weaker than the topology r 2 if r 1 c r ‘1.
Springer : Review: Lars V. Ahlfors and Leo Sario, Riemann surfaces
Mii1Nr of which ia tloitl. Sario – Riemann Surfaces Alexandre row Enviado por: This ahlforss applies equally well to Oz, and we find that 01 and 02 cannot both be nonempty and at the same time disjoint. It must be observed, however, that the new space is not necessarily a Hausdorff space even if Sis one.
The space obtained by identification can be referred to as the qvotientspau of S with respect to the equivalence relation whose equivalence classes are the sets P. It is therefore convenient to eiemann the notion of a basis for the open sets briefly: B The intersection of any finite collldion of sets in fJI is a union of sets in We shall say that a family of closed seta has the finite inter- M. The empty set and the Whole apace are simultaneously open and closed. For instance, it cannot be proved by analytical means that every surface which satisfies the axiom of countability can be made into a Riemann surface.
Lars V. Ahlfors, L. Sario – Riemann Surfaces
For the points of the plane Jl2 we shall frequently use the complex notation The sphere 81, also referred to as the u: Since we strive for completeness, a considerable part of the first chapter has been allotted to the oombinatorial approach. A compact subset is of course one which is compact in the relative topology.
AI The union of any collldion of open sets is open. We shall always understand that the topology on a subset is its relative topology. Tbia is to be contrasted with ahlfkrs formula AnBcAnB which is weaker inasmuch as it gives only an inclusion.
It so happens that this superficial knowledge is adequate for most applications to the theory of Riemann surfaces, and our presentation is influenced by this fact. Mostly, we consider only neighborhoods of points, and we use the notation V p to indicate that Vis a neigj borhood ofp.
As a consequence of 3B every point in a topological space belongs to a maximal connected subset, namely the union of all connected sets which contnin the given point. Uon of a finite oolleotion of closed seta are closed. But 01 is also relatively closed in Q.
Lars V. Ahlfors, L. Sario – Riemann Surfaces – livro em pdf
A2 The intersection of any finite coUection of open sets is open. The open and closed seta on S’ are often referred to as relatively open or closed sets. We have tried, however, to isolate this pq. This is the moat useful form of the definition for a whole category of proofs. This shows that 0 is open. The fundamental group is introduced, and the notion of bordE.
The boundary of P is formed by all points which belong neither to the interior nor to the surfaxes. A closed connected set with more than riiemann point is a conlinuum. Such a basis is a system fJI of subsets of 8 which satisfies condition.