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 exists, for it can be obtained as t lw intersection of all closed sets which contain P.

For the points of the plane Jl2 we shall frequently use the complex notation The sphere 81, also referred to as the u: The main demerit of this approach is that it does not yield complete results. We call this topology on S’ the relative topology induced by the topology on B. The requirements are interpreted to hold also for the empty collection. The empty set and all seta with only one point are trivially ahltors nected. This property neither implies nor is implied by connectedness in the large.

A Migllborlwod of a set Rkemann c 8 is a set V c 8 which.

## Lars V. Ahlfors, L. Sario-Riemann Surfaces

Durfaces construction of a quotient-apace is sometimes preceded by forming the sum of several spaces. Mii1Nr of which ia tloitl. This is done by formulating the combinatorial theory as a theory of triangulated surfaces, or polyhedrons. 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. The strongest topology is the tliscrete topology in which every subset is open. Sometimes it is more convenient to express compactneea in terms of closed seta.

A compact subset is of course one which is compact iremann the relative topology.

### Lars V. Ahlfors, L. Sario – Riemann Surfaces – livro em pdf

A2 The intersection of any finite coUection of open sets is open. Examples of this type of proof will be abundant. It has been found most convenient to base the definition on the consideration of open coverings.

Surface nano-architecture of a metalorganic framework Surface nano-architecture of a metalorganic framework. Sqrio, this is equivalent to saying that the family of complements contains no finite covering. In most a third requirement is added:. There is a great temptation to bypass the finer deta. Denote the given seta by P.

A topology 9″ 1 is said to be weaker than the topology r 2 if r 1 c r ‘1. 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.

Surface acetylation of bacterial cellulose Surface acetylation of bacterial cellulose. The boundary of P is formed by all points which belong neither to the interior nor to the exterior. This shows that 0 is open.

According to usual conventions the union of an empty collection of sets is the empty set 0, and the intersection of an empty collection is the whole space 8. Ction properly if any finite number of seta in the family have a non void interaection.

## Lars V. Ahlfors, L. Sario – Riemann Surfaces

The definition applies also to subsets in their relative topology, and riiemann can hence apeak of connected and disconnected subsets. Tbia is to be contrasted with the formula AnBcAnB which is weaker inasmuch as it gives only an inclusion.

Since we strive for completeness, surfwces considerable part of the first chapter has been allotted to the oombinatorial approach. Such a basis is a system fJI of subsets of 8 which satisfies condition B The intersection of any finite collldion of sets in fJI is a union of sets in We have tried, however, to isolate this pq. A topology in the space of points P can be introduced by the agreement that sirfaces set is open if and only if the union of the corresponding seta Pis an open set in B.