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Research report: "the probability measure functor preserving some topological properties that"

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Collection of research reports best university in 2008 honored the author: 1. Housing Ta Khac Luong Quoc Tuyen Nguyen Thi Thu Ha, Le Thi Ngoc, the probability measure functor preserving some topological properties that . science (Scientia, in Latin, meaning "knowledge" or "understanding") is the efforts to implement the invention, and increased knowledge of the human understanding of how the operation of the physical world around them. Through controlled methods, scientists use to observe the signs of expression or of the material and unusual nature to collect data, analyze information to explain how it works, there at the phenomenon of. | PROBABILITY MEASURE FUNCTORS PRESERVING SOME TOPOLOGICAL PROPERTIES TA KHAC cu a LUONG QUOC TUYEN b NGUYEN THI THU HA c LE THI NGOC c Abstract. In this paper we introduce a topology in Pk X and prove th at if X have some networks then Pk X so have. 1. Introduction AND NOTIONS In 1 V. V. Fedorchuk introduced the concept of probability measure functors with finite supports and proved that these functors preserve some topological invariants. In this paper we study the action of probability measure functors on space with networks and show some invariants as networks cs- networks cs -networks . under probability measure functors. Definition 1.1. Let X be a Hausdorff space. A probability measure with finite supports on X is a function y X 0 1 satisfying the conditions 1 suppy x E X y x 0 is finite 2 E A x 1. xesupPm For every k E N let Pk X denote the set of all probability measures onX whose supports of no more than k points. Then each y E Pk X can be written in the form x . f0 if y x y miỗxi with q k where Sx y is Dirac function that is Sx y 2 1 Ự if y x q and mi y xi 0 mi 1. mi y xi is called the mass of y at xi. In 1 i 1 Fedorchuk introduced a topology on Pk X as follows Let y E Pk X y y q 1 mi JXi. A neighbor ho od of y is a set in the form O y Ui U2 . Uq E q 1 y E Pk X y y2 yi SUPP yi c Ui III yi II -mi E i 1 2 . q 1 mq i 0 i 1 1 Nhận bài ngày 08 5 2007. sửa chữa xong ngày 20 10 2007. where E 0 U U2 . Uq are sjomt neighborhoods of xi x2 . xq respectively Uq i X uq i X. and II ụ 11 ụ x - xesup Pm t Ui Note that ụ is not necessarily a measure i 1 2 . q 1 because in general II ụ 11 1 Let B be a base of the topology in X. Then the family O ụ Ui . Uq e ụ E Pk X Ui E B i 1 2 . q q k E 0 forms a base of a topology on Pk X . This topology is called Fedorchuk topology. qq Definition 1.2. Let ụ miỏxi E Pk X ụn y2 m dxv- q c k. The sequence i 1 i 1 ụn is called convergent to ụo iff for E 0 there exists N 0 such that for every n N we have xn E Ui i 1 . q and mrn m0 E i 1 . q.