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Minimal Hausdorf Spaces | ||
| Journal of Kufa for Mathematics and Computer | ||
| Article 1, Volume 1, Issue 4, 2013, Pages 116-119 | ||
| Authors | ||
| Hadi J. Mustafa; Mohammed S. Mechee | ||
| 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 : 1-1 definition :[4] let be a space , we say that X is a minimal space if implies that is not space . 2. Main Results In this section , we proved several properties of minimal spaces 2-1 Theorem The property of being minimal is a topological property Proof : Let be a minimal space Let be Homeomorphism from a minimal onto a topological space , The topological space is because the property of being is a topological property Now assume that is not a minimal space , this means that there exists a topology such that is a space Let then is a topology on X such that We will show that is a space Let , Consider , | ||
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