Minimal Hausdorf Spaces

Abstract

Abstract In this work , we study minimal Hausdorf spaces , these are Hausdorf spaces where implies that is not Hausdorf space , Several theorems about this concept have been proved . 1.Introduction Let be a Topological space , . if is a Hausdorf spaces ( space) the is not necessarily , for example : is space( is usual topology on the set of all real numbers R ) but is not ( is the co finite topology on R )This motivates the definition of minimal Hausdorf spaces as :1-1definition :[4] let be a space , we say that X is a minimal space if implies that is not space .2.Main ResultsIn this section , we proved several properties of minimal spaces 2-1 TheoremThe property of being minimal is a topological propertyProof :Let be a minimal spaceLet be Homeomorphism from a minimal onto a topological space , The topological space is because the property of being is a topological propertyNow assume that is not a minimal space , this means that there exists a topology such that is a spaceLet then is a topology on X such that We will show that is a space Let , Consider ,