TAILIEUCHUNG - Research report: "Several have commented on the tight space count."

Specifically, the science of physics to the laws of natural movement, from a macroscopic (particles that make up matter) to larger scales (planets, galaxies and the universe). In English, from physics (physics) is derived from Greek φύσις (phusis) means natural and φυσικός (phusikos) is of the nature. The main object of study now include physical matter, energy, space and time. | NOTES ON COUNTABLE TIGHTNESS SPACES NGUYEN VAN DUNG a Abstract. In this paper we introduced the definition of countable closure operators and use it to characterize countable tightness spaces. Moreover we consider the generalizations of some results of w. c. Hong in 4 5 . 1. Introduction Let X be a set P X the family of all sub sets of X S X the set of all sequences of points in X N the set of all natural numbers and R the set of all real numbers. Let X c be a topological space endowed with the closure operator c . A function c P X P X where c A x G X xn x for some sequence xn G S A for every A c X is called a sequential closure operator 5 . The sequential closure operator c on a topological space X c possesses the following properties . Lemma. 5 Let X c be a topological space with the closure operator c. 1 c 2 A c c A c c A for every A c X 3 If A c B c X then c A c c B 4 c A u B c A u c B for every A B c X. In other way c possesses the Kuratowski closure axoms except for idempotent. So it need not be a topological closure operator on the set X. Recall the following definitions in a topological space X c endowed with the closure operator c. . Definition. Let X be a topological space. 1 X is called a sequential space 1 if for each subset A of X A is closed provided that if xn G S A and xn converges to x then x G A. 2 X is called a Frechet space 1 if for each subset A of X and any point x G c A there exists some sequence xn c A such that xn converges to x. 1 - 2000 Mathematics Subject Classification. 54A20 54D35 54D55 54D80 55E25. - Keywords. sequential Frechet weakly first countable countable tightness closure operator sequential closure operator countable closure operator. - Received 10 01 2006 in revised form 25 02 2006. 3 X is called a countable tightness space or a space X has countable tightness 1 if for each subset A of X A is closed provided that c C c A for every countable subset C of A. 4 X is called a weakly first countable space 5 if for each x G X

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