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

Evidently, this is equivalent to saying that the family of complements contains no finite covering. It has been found most convenient to base the definition on the consideration of open coverings.

As a consequence of 3B every point in aglfors topological space belongs to a maximal connected subset, namely the union of all connected sets which contnin the given point. If 01 is not empty it meets at least one 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. We shall always understand that the topology on a subset is its relative topology.

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. This property neither implies nor is implied by connectedness in saario large. We have tried, however, to isolate this pq. C C X Xcomo In most a third rie,ann is added: AI The union of any collldion of open sets is open. We proceed to the definition of compact spaces.

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

Mostly, we consider only neighborhoods of rieman, and we use the notation V p to indicate that Vis a neigj borhood ofp. In most a third requirement is added:. Sario – Riemann Surfaces Alexandre row Enviado por: The empty set and all seta with only one point are trivially con- nected. It is a Hausdorfl” topology 1f and only if any two distinct points belong to disjoint sets in It so happens ahlforrs this superficial knowledge is adequate for most applications to the theory of Riemann surfaces, and our presentation is influenced by this fact.

A2 The intersection of any finite coUection of open sets is open.

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

Since we strive surfqces completeness, a considerable part of the first chapter has been allotted to the oombinatorial approach. Suppose that p belongs to the component 0, and let V p be a connected neighborhood. 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. From Q c 01 it follows that Oa is empty, and hence Q is connected. A compact subset is of course one which is compact in the relative topology.

For complete results this derivation must be based on the method of triangulation.

A apace with more than one point can be topologized in different manners. Such a basis is a system fJI of subsets of 8 which satisfies condition. It shows that every open set in a locally connected space is a union of disjoint regions.

If thiH is so, one of the seta must be empty, and we conclude that the property holds for all points or for no points. We shall say that a family of closed seta has the finite inter- M. For this reason sufraces topological theory of surfaces belongs in this book. For instance, a compact space can thus be covered by a finite saro of from an surfcaes basis.

They are relatively open with respect to P. A space iB compact if and only if every open covering contains a finite BUbcovering. It exists, for it can be obtained as t lw intersection of all closed sets which contain P. But 01 is also relatively closed in Q.

The compomntB of a locally connuted space are Bimtdtamously open and cloBed. The following theorem is thus merely a rephrasing of the definition. 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. One of these is the process of relativization. The topological product 81 x Sa is defined as follows: An open covering of a subset is then a family of open seta ahofors union contains the given subset, and if the subset is compact we can again select a finite aubcovering.