%0 Journal Article
%T Study of a Forwarding Chain in the Category of Topological Spaces between and with respect to One Point Compactification Operator
%A Fatemah Ayatollah Zadeh Shirazi
%A Meysam Miralaei
%A Fariba Zeinal Zadeh Farhadi
%J Chinese Journal of Mathematics
%D 2014
%R 10.1155/2014/541538
%X In the following text, we want to study the behavior of one point compactification operator in the chain := Metrizable, Normal, , KC, SC, US, , , , , Top of subcategories of category of topological spaces, Top (where we denote the subcategory of Top, containing all topological spaces with property , simply by ). Actually we want to know, for and , the one point compactification of topological space belongs to which elements of . Finally we find out that the chain Metrizable, , KC, SC, US, T1, , , , Top is a forwarding chain with respect to one point compactification operator. 1. Introduction The concept of forwarding and backwarding chains in a category with respect to a given operator has been introduced for the first time in [1] by the first author. The matter has been motivated by the following sentences in [1]: ¡°In many problems, mathematicians search for theorems with weaker conditions or for examples with stronger conditions. In other words they work in a subcategory of a mathematical category, namely, , and they want to change the domain of their activity (theorem, counterexample, etc.) to another subcategory of like such that or according to their need.¡± Most of us have the memory of a theorem and the following question of our professors: ¡°Is the theorem valid with weaker conditions for hypothesis or stronger conditions for result?¡± The concept of forwarding, backwarding, or stationary chains of subcategories of a category tries to describe this phenomenon. In this text, Top denotes the category of topological spaces. Whenever is a topological property, we denote the subcategory of Top containing all the topological spaces with property , simply by . For example, we denote the category of all metrizable spaces by Metrizable. We want to study the chain {Metrizable, Normal, , KC, SC, US, , TD, TUD, , Top} of subcategories of Top in the point of view of forwarding, backwarding, and stationary chains¡¯ concept with respect to one point compactification or Alexandroff compactification operator. Remark 1. Suppose is a partial order on . We call (i)a chain, if for all , we have ;(ii)cofinal, if for all , there exists such that . In the following text, by a chain of subcategories of category , we mean a chain under ¡° ¡± relation (of subclasses of ). We recall that if is a chain of subcategories of category such that is closed under (multivalued) operator , then we call (i)a forwarding chain with respect to ; if for all , we have (i.e., );(ii)a full-forwarding chain with respect to ; if it is a forwarding chain with respect to and for all distinct , we
%U http://www.hindawi.com/journals/cjm/2014/541538/