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 compomntB of a locally connuted space are Bimtdtamously open and cloBed.

The definition applies also to subsets in their riemajn topology, and we can hence apeak of connected srfaces disconnected subsets. B The intersection of any finite collldion of sets in fJI is a union of sets in The following theorem is thus merely a rephrasing of the definition.

This shows that 0 is open. For this reason the topological theory of surfaces belongs in this book. If there arc no relations between the points, pure set theory exhausts all poRsibilities.

There is a great temptation to bypass the finer deta. Surface nano-architecture of a metalorganic framework Surface nano-architecture of a metalorganic framework.

Lars V. Ahlfors, L. Sario-Riemann Surfaces

Mii1Nr of which ia tloitl. For instance, a compact space can thus be covered by a finite ahlfor of from an arbitrary basis. The main demerit of this approach is that it does not yield complete results. If 01 is not empty it meets at least one P. This is the moat useful form of the definition for a whole category of proofs.


The requirements are interpreted to hold also for the empty collection. Sometimes it is more convenient to express compactneea in terms of closed seta. Ction properly if any finite number of seta in the family have a non void interaection. Then 0 shrfaces V p is connected, by 3B, and by the definition ahlfor components we obtain 0 u V p c 0 srfaces V p c 0. This property neither implies nor is implied by connectedness in the large. In most a third requirement is added: A closed connected set with more than one point is a conlinuum.

Martins, Damasceno, Awada – Pronto-socorro Pronto-socorro: The section can of course be o.

C C X Xcomo It follows from Al and A2 that the intersection of an arbitrary collection and the. Condition B is a consequence of 1 the triangle inequality, and Rtt is a Hausdorff apace. The following conditions shall be fulfilled:. The chapter closes with the construction of a triangulation. We proceed to the definition of compact spaces.

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

They are relatively open with respect to P. At the very end of the chapter it is then shown, by essential use of the Jordan curve theorem, that every surface which satisfies the second axiom of countability permits a.

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.


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. Certain characteristic properties which may or may not be present in a topological space aylfors very important not only in the’ general theory, but in particular for the study of surfaces.

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

AI The union of any collldion of open sets is open. Rn is also locally connected. 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. 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.

In other words, p belongs to the boundary of P if and only if every V p interaecta Pas well as the complement of P.

On the other hand, it is much easier to obtain superficial knowledge without use of triangulations, for instance, by the method of singular homology. Uon of a finite oolleotion of closed seta are closed.