فضاهاي حل شدني

چكيده انگليسي :

Resolvable Spaces Narges Javani n javani@math iut ac ir December 2016 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Mohammad Reza Koushesh koushesh@cc iut ac irAdvisor Dr Majid Gazor mgazor@cc iut ac ir2015 MSC Primary 54A05 54C30 Secondary 54C99 54D05 54D10Keywords Resolvable Irresolvable Locally Connected Connected Hausdorff Functionally Haus dorffAbstract Let X X be a topological space and let X be the union of two respectively disjoint densesubsets then X is called resolvable resolvable respectively A space which is not resolvable not resolvable respectively is called irresolvable irresolvable respectively The aim of present the sis is to determine conditions under which a space is a resolvable space This thesis is divided into fivechapters In the first chapter we mention required definitions and theorems The methods needed tostudy irresolvable spaces are those needed to study expansion of spaces For this reason the secondchapter is devoted to the development of the theory of such expansions We investigate properties ofspaces which are preserved under arbitrary expansions such as the property of being a T0 space T1space Hausdorff space and Urysohn space In considering various types of expansions we prove theexistence of a particular expansion called a maximal expansion enjoying a number of certain specialproperties Maximal space is essential in the construction of irresolvable spaces Also we consider vari ous properties of contraction which is the reverse operation with respect to expansion The third chaptercontains a study of irresolvable spaces The existence of totally disconnected Urysohn spaces irresolv able connected T1 spaces and totally disconnected completely regular spaces are proved directly fromthe expansion theory Some irresolvable spaces enjoy a very strong disconnectivity property which weinvestigate in detail In the fourth chapter it is shown that metric spaces compact Hausdorff spaces Hausdorff spaces satisfying the first countability axiom and various other special spaces are all resolv able We also obtain a reduction showing that only T1 spaces and in fact only compact T1 spaces are needed to be considered in dealing with the resolution problem In the final chapter we study under whatconditions a locally connected space and DG space can be resolved that is are the union of disjointsubsets each of which is dense in the space under consideration In this chapter we use the notion ofretodic spaces to obtain a sufficient condition for resolvability of locally connected spaces For acardinal we say that a T1 space Y is a retodic space if Y is partitioned into a family D kof dense subsets in such a way that the complement of every D is totally disconnected It is shownthat if for every connected open subset U of X there exists a retodic space YU and a non constantcontinuous function f U YU then a locally connected space X is resolvable As a result every locally connected functionally Hausdorff space is resolvable and also every irresolvable lo cally connected T1 space contains an open connected subset in which all continuous functions on it toany retodic space all real continuous functions on it are constant A topological space X is said tobe a DG space if every subset of X is a G set in its closure We consider the relationship betweenresolvable and irresolvable spaces with DG space We also consider some new results in these classesof topological spaces

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فضاهاي حل شدني

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