Study of a Forwarding Chain in the Category of Topological Spaces between and with respect to One Point Compactification Operator

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