TAILIEUCHUNG - Báo cáo " On the matheron theorem for topological spaces"

In this paper we study the extending of the Matheron theorem for general topological spaces. We also show some examples about the spaces F such that the miss-and-hit topology on those spaces are unseparated or non-Hausdorff. 1. Introduction The Choquet theorem (see [1, 2]) plays very importance role in theory of random sets. The proof of this theorem is based on the Matheron theorem and especially, the locally compact property of the space F , where F is a space of all close subsets of a given space E and F is equipped with the miss-and-hit topology (see [1]). . | VNU Journal of Science Mathematics - Physics 23 2007 194-200 On the matheron theorem for topological spaces Dau The Cap1 Bui Dinh Thang2 1 Hochiminh city University of Pedagogy 280 An Duong Vuong Dist 5 Hochiminh city Vietnam Saigon University 273 An Duong Vuong Dist 5 Hochiminh city Vietnam Received 15 September 2007 received in revised form 1 November 2007 Abstract. In this paper we study the extending of the Matheron theorem for general topological spaces. We also show some examples about the spaces F such that the miss-and-hit topology on those spaces are unseparated or non-Hausdorff. 1. Introduction The Choquet theorem see 1 2 plays very importance role in theory of random sets. The proof of this theorem is based on the Matheron theorem and especially the locally compact property of the space F where F is a space of all close subsets of a given space E and F is equipped with the miss-and-hit topology see 1 . The Matheron theorem is stated as follows. Theorem. Let E be a complete separable and locally compact metric space. Then the miss-and-hit topology on F space of all closed subsets of E is compact separable and Hausdorff Note that the natural domain of the probability theory is a Polish space which is in general not locally compact. So in 3 the authors extended the Matheron theorem for general metric space. They showed that if E is a separable metric space then the miss-and-hit topology on space F is separable and compact. And if E has a non-locally compact point then the miss-and-hit topology on space F is not Hausdorff. Now we extend the Matheron theorem for general topological space. Let E be a topological space. Denote F K and G the families of all close compact and open subsets of E respectively. For every A c E we denote Fa F F eF F n A 0 FA F F eF F n A 0 . For every K E K and a finite family of sets G1 . Gn eg n E N we put FK G. F K n F. n Fc . Then F- . Gn K EK Gi . Gn eg n E N is a base of topology on F. Which is called a miss-and-hit topology on

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