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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The chapter closes with the construction of a triangulation.

C C X Xcomo From Q c 01 it follows that Oa is empty, and hence Q is connected. This definition has an obvious generalization to the case of an arbi- trary collection of topological spaces.

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

The ssrio demerit of this approach is that it does not yield complete results. We shall say that a family of closed seta has the finite inter- M.

On the other hand, it is much easier to obtain superficial knowledge without use of triangulations, for instance, by the method of singular homology.

The topological product 81 x Sa is defined as follows: The empty set and the Whole apace are simultaneously open and closed. One of these is the process of relativization. Tbia is to be contrasted with the formula AnBcAnB which is weaker inasmuch as it gives only an inclusion. The requirements are interpreted to hold also for the empty collection. An open connected set is called a region, and the closure of a region is referred to as a cloM.

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

The following conditions shall be fulfilled:. We proceed to the definition of compact spaces. Suppose that p belongs to the component 0, and let V p be a connected neighborhood. A2 The intersection of any finite coUection of open sets is open. Sario – Riemann Surfaces Alexandre row Enviado por: Evidently, this is equivalent to saying that the family of complements contains no finite covering.

AI The union of any collldion of open sets is open. B The intersection of any finite collldion of sets in fJI is a union of sets in 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. The construction of a quotient-apace is sometimes preceded by forming the sum of several spaces. The definition applies also to subsets in their relative topology, and we can hence apeak of connected and disconnected subsets.

This shows that 0 is open.

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

It follows from Al and A2 that the intersection of an arbitrary collection and the. The complement of an open set is said to be cloftd. 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. The strongest topology is the tliscrete topology in which every surtaces is open.

A topology 9″ 1 is said to be weaker than the topology r 2 if r 1 c r ‘1. An open covering of a subset is then a family of open seta whose union contains the given subset, and if the subset is compact we can again select a finite aubcovering.

The empty set rjemann all seta xurfaces only one point are trivially con- nected. From the connectedness of P. Rn is also locally connected. This reasoning applies equally well to Oz, and we find that 01 and 02 cannot both be nonempty and at the same time disjoint.

It is a Hausdorfl” topology 1f and only if any two distinct points belong to disjoint sets in A topology in the space of points P can be introduced by the agreement that a set is open if and only if the union of the corresponding seta Pis an open set in B.

Ction properly if any finite number of seta in the family have a non void interaection. A compact subset is of course one which is compact in the relative topology.