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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In most a third requirement is added:. 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 It shows that every open set in a locally connected space is a union of disjoint regions. This shows that 0 is open.
Riemann Surfaces – Lars V. Ahlfors, Leo Sario – Google Books
This means that points can be identified which were not initially in the same space. This reasoning applies equally well to Oz, and we find that 01 and 02 cannot both be nonempty and at the same time disjoint. 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.
A Riemann surface is, in the first place, a surface, and its properties depend to a very great extent on the topological character of the surface. We shall say that a family of closed seta has the finite inter- M. The following theorem is thus merely a rephrasing of the definition. From Q c 01 it follows that Oa is empty, and hence Q is connected. The boundary of P is formed by sadio points which belong neither to the interior nor to the exterior. Mii1Nr of which ia tloitl. Since we strive for completeness, a considerable part of the first chapter has been allotted to the oombinatorial approach.
It follows from Al and A2 that the intersection of an arbitrary collection and the. One of these is the process of relativization. From the connectedness of P. 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. But 01 surfacs also relatively closed in Q.
If there arc no relations between the points, pure set theory exhausts all poRsibilities. It is a Hausdorfl” topology 1f and only if any two distinct points belong to disjoint sets in A safio connected set with more than one point is a conlinuum.
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. C C X Xcomo We call this topology on S’ the relative topology induced by the topology on B. We proceed to the definition of compact spaces. 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.
Examples of this type of proof will be abundant. Suppose that p belongs to the component 0, and let V p be a connected neighborhood. An open connected set is called a region, and the closure of a region is referred to as a cloM. Uon of a finite oolleotion of closed seta are closed. 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. A2 The intersection of any finite coUection of open sets is open.
For complete results this derivation must be based on the method of triangulation.
Springer : Review: Lars V. Ahlfors and Leo Sario, Riemann surfaces
For instance, it cannot be proved by analytical means xhlfors every surface which satisfies the axiom of countability can be made into a Riemann surface. Sario – Riemann Surfaces Alexandre row Enviado por: B The intersection of any finite collldion of sets in fJI is a union of sets in Mostly, we consider only neighborhoods of points, and we use zhlfors notation V p to indicate that Vis a neigj borhood ofp.
Evidently, this is equivalent to saying that the family of complements contains no finite covering. A compact subset is of course one which is compact in ajlfors relative topology.
The empty set and all seta with only one point are trivially con- nected. In other instances the analytical method becomes so involved that it no longer possesses the merit of elegance.
Hence 01 contains the l’ommon point of all Pa. This is the moat useful form suefaces the definition for a whole category of proofs.
Lars V. Ahlfors, L. Sario – Riemann Surfaces
Again, AlHA2 are trivially fulfilled. The topological product 81 x Sa is defined as follows: It has been found most convenient to riemamn the definition on the consideration of open coverings.
If 01 is not empty it meets at least one P. The section can of course be o.