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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A2 The intersection of any finite coUection of open sets is open. The following conditions shall be fulfilled: Sometimes it is more convenient to express compactneea in terms of closed seta. This shows that 0 is open.

It must be observed, however, that the new space is not necessarily a Hausdorff space even if Sis one. An open set riwmann a neighborhood of any subset, and a set is open if it contains a neighborhood of every point in the set. Denote the given seta by P.

Such a basis is a system fJI of subsets of 8 which satisfies condition. The boundary of P is formed by all points which belong neither to the interior nor to the exterior.

### Lars V. Ahlfors, L. Sario-Riemann Surfaces – PDF Drive

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. Uon of a finite oolleotion of closed seta are closed.

But 01 is also relatively closed in Q. Rn is also locally connected. This property neither implies nor is implied by connectedness in the large. Surface acetylation of bacterial cellulose Surface acetylation of bacterial cellulose.

From Q c 01 it follows that Oa is empty, and hence Q is connected. It shows that every open set in a locally connected space is a union of disjoint regions. According to usual conventions the union of an empty collection of sets is the halfors set surfsces, and the intersection of an empty collection is the whole space 8.

If there arc no relations between the points, pure set theory exhausts all poRsibilities. This maximal connected set is called the romJmnPnl of the “l”H’I’ dt’l’l’llined hy the point.

The empty set and all seta with only one point are trivially con- nected. A topology in the space of points P can be introduced by the agreement that a set is open if and only if surfaes union of the corresponding seta Pis an open set in B.

For instance, it cannot be proved by analytical means that every surface which satisfies the axiom of countability can be made into a Riemann surface. It is a Hausdorfl” topology 1f and only if any two distinct points belong to disjoint sets in In fact, Al – A2 rimann ouvwusly satisfied; A3 holds if it holds on B. It follows from Al and A2 that the intersection of an arbitrary collection and the.

Suppose that p belongs to the component 0, and let V p be a connected neighborhood. We call this topology on S’ the relative giemann induced by the topology on B. C C X Xcomo In the case of a aubaet it is convenient to replace the relatively open seta of a covering by corresponding open seta of the whole apace. The sum of two topological spaces 81, Sa is their union 81 u Bz on which the open seta are those whose intersections with 81 and Sa are both open.

The following theorem is thus merely a rephrasing of the definition. It so happens that this superficial knowledge is ahlgors for most applications to the theory of Riemann surfaces, and riemqnn presentation is influenced by this fact. If 01 is not empty it meets at least one P.

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

For instance, a compact space can thus be covered by a finite number of from an arbitrary basis. A space iB compact ahlfos and only if every open covering contains a finite BUbcovering. Surface nano-architecture of a metalorganic framework Surface nano-architecture of a metalorganic framework. The fundamental group is introduced, and the notion of bordE. One of these is the process of relativization.

ahlfor The topological product 81 x Sa is defined as follows: 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. The following conditions shall be fulfilled:.