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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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. Examples of this type of proof will be abundant.

Condition B is a consequence of 1 the triangle inequality, and Rtt is a Hausdorff apace. An open covering is a family of open sets whose union is the whole space, and a covering is finite if the family contains only a finite number of sets.

Riemann Surfaces – Lars V. Ahlfors, Leo Sario – Google Books

This definition has an obvious generalization to the case of an arbi- trary collection of topological spaces. C C X Xcomo 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. The following theorem is thus merely saio rephrasing of the definition. A Migllborlwod of a set A c 8 is a set V c 8 which.


The empty set and all rlemann with only one point are trivially con- nected.

Lars V. Ahlfors, L. Sario – Riemann Surfaces

The section can of course be o. From the connectedness of P. Hence 01 contains the l’ommon point of all Pa.

The complement of an open set is surfacfs to be cloftd. This elementary section has been included for the sakP of l”ompleteness and because beginning analysts are not always well prepared on this point.

Lars V. Ahlfors, L. Sario-Riemann Surfaces

Every open subset of a locally connected space is itself locally connected, for it has a basis conaisting of part of the basis for the whole space. It has been roemann most convenient to base the definition on the consideration of open coverings.

Rn is also locally connected.

It follows from Al and A2 that the intersection of an arbitrary collection and the. In other instances the analytical method becomes so involved that it no longer possesses the merit of elegance. Denote the given seta by P. The topological product 81 x Sa is defined as follows: 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.

Then 0 u V p is connected, by 3B, and by the definition of components we obtain 0 u V p c 0 or V p c 0. Since we strive for completeness, a considerable part of the first chapter has been allotted to the oombinatorial approach.


From Q c 01 it follows that Oa is empty, and hence Q is connected. We shall say that a family of closed seta has the finite inter- M. This means that points can be identified which were not initially in the same space. The definition applies also to subsets in their relative topology, and we can hence apeak of connected and disconnected subsets.

This maximal connected set is called the romJmnPnl of the “l”H’I’ dt’l’l’llined hy the point.

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

There is a great temptation to bypass the finer deta. It is a Hausdorfl” topology 1f and only if any two distinct points belong to disjoint sets in A space ill locally con? For the points of the plane Jl2 we shall frequently use the complex notation The sphere 81, also referred to as the u: 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.