MATRIKS KETERHUBUNGAN LANGSUNG TOPOLOGI HINGGA

Abtrak: Topologi merupakan cabang ilmu matematika yang mempelajari suatu struktur yang terdapat pada himpunan. Seperti halnya himpunan hingga yang memiliki kardinalitas, maka topologi hingga juga memiliki kardinalitas. Jika himpunan memiliki kardinalitas dan topologi pada S, maka kardinalitas dari yang dinotasikan dengan menyetakan banyaknya elemen dari . Jika topologi pada S, maka matriks keterhubungan langsung topologi adalah matriks berukuran yang dinotasikan dengan . Matriks merupakan matriks yang elemennya 0 atau 1. Kata Kunci: Himpunan, Kardinalitas, Matriks Keterhubungan Langsung, Topologi. Abstract: Topology is a branch of mathematics which study structures on a set. As a finite set, a finite topology have a cardinality. Let be a finite set with cardinality and let be a topology on S, then the cardinality of which denotes is the number of elements . If topology on S, then the corresponding matrix to a topology is a matrix which denoted by . is the matrix have element 0 or 1. Keywords: Cardinality, Set, The Corresponding Matrix, Topology,

Embedded minimal surfaces of finite topology

In this paper we prove that a complete, embedded minimal surface M in {\mathbb{R}^{3}} with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface {\overline{M}} with boundary punctured in a finite number of interior points and that M can be represented in terms of meromorphic data on its conformal completion {\overline{M}}. In particular, we demonstrate that M is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior of M.

A construction of complete complex hypersurfaces in the ball with control on the topology

Given a closed complex hypersurface {Z\subset\mathbb{C}^{N+1}} ({N\in\mathbb{N}}) and a compact subset {K\subset Z}, we prove the existence of a pseudoconvex Runge domain D in Z such that {K\subset D} and there is a complete proper holomorphic embedding from D into the unit ball of {\mathbb{C}^{N+1}}. For {N=1}, we derive the existence of complete properly embedded complex curves in the unit ball of {\mathbb{C}^{2}}, with arbitrarily prescribed finite topology. In particular, there exist complete proper holomorphic embeddings of the unit disc {\mathbb{D}\subset\mathbb{C}} into the unit ball of {\mathbb{C}^{2}}. These are the first known examples of complete bounded embedded complex hypersurfaces in {\mathbb{C}^{N+1}} with any control on the topology.

On characterizations of finite topological spaces with granulation and evidence theory

The theory of finite topological spaces can be used to investigate deep well-known problems in Topology, Algebra, Geometry and Artificial Intelligence. To represent uncertainty knowledge of a finite topological space, two kinds of measurement of a finite topological space are first introduced. Firstly, a kind of granularity of a finite topological space is defined, and properties of the granularity are explored. Secondly, relationships between the belief and plausibility functions in the Dempser-Shafer theory of evidence and the interior and closure operators in topological theory are established. The probabilities of interior and closure of sets construct a pair of belief and plausibility functions and its belief structure. And, for a belief structure with some properties, there exists a probability and a finite topology such that the belief and plausibility functions defined by the given belief structure are, respectively, the belief and plausibility functions by the topology. Then a necessary and sufficient condition for a belief structure to be the belief structure induced by a finite topology is presented.

Completely simple endomorphism rings of modules

<p>It is proved that if A_p is a countable elementary abelian p-group, then: (i) The ring End (A_p) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End (A_p)/I, where I is the ideal of End (A_p) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End (A_p) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of modules over commutative rings is obtained.</p>