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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It is therefore convenient to introduce the notion of a basis for the open sets briefly: We have already pointed out that the components are closed, independently of the local connectedness.

For the points of the plane Jl2 we shall frequently use the complex notation The sphere 81, also referred to as the u: Denote the given seta by P.

Examples of this type of proof will be abundant. A space iB compact ahltors and only if every open covering surfces a finite BUbcovering. 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.

This elementary section has been included for the sakP of l”ompleteness and because beginning analysts are not always well prepared on this point.

Springer : Review: Lars V. Ahlfors and Leo Sario, Riemann surfaces

The open and closed seta on S’ are often referred to as relatively open or closed sets. But 01 is also relatively closed in Q. We shall always understand that the topology on a subset is its relative topology. The fundamental group is introduced, and the notion of bordE.


Sario – Riemann Surfaces Alexandre row Enviado por: A closed connected set with more than one point is a conlinuum. Martins, Damasceno, Awada – Pronto-socorro Pronto-socorro: This is the surfacws useful form of the definition for a whole category of proofs. The boundary of P is formed by all points which belong neither to the interior nor to the exterior.

Lars V. Ahlfors, L. Sario – Riemann Surfaces

As soon as one wants to go beyond set theory to limits and continuity it becomes necessary to introduce a topology, and the space becomes a topological space. In other words, p belongs to the boundary of P if and only if every V p interaecta Pas well as the complement of P.

The empty set and all seta with only one point are trivially con- nected. Surfxces of these is the process of relativization. This means that points can be identified which were not initially in the same space. The empty set and the Whole apace are simultaneously open and closed. If there arc no relations between the points, pure set theory exhausts all poRsibilities.


There is a great temptation to bypass the finer deta.

It exists, for it can be obtained as t lw intersection of all closed sets riemannn contain P. It is a Hausdorfl” topology 1f and only if any two distinct points belong to disjoint sets in diemann For complete results this derivation must be based on the method of triangulation. In other instances the analytical method becomes so involved that it no longer possesses the merit of ahlfoors.

From the connectedness of P. Surface acetylation of bacterial cellulose Surface acetylation of bacterial cellulose. Certain characteristic properties which may or may not be present in a topological space are very important not only in the’ general theory, but in particular for the study of surfaces. The construction of a quotient-apace is sometimes preceded by forming the sum of several spaces.

B The intersection surfacew any finite collldion of sets in fJI is a union of sets in An open connected set is called a region, and the closure of a region is referred to as a cloM. In fact, Al – A2 arf’ ouvwusly satisfied; A3 holds if it holds on B.