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Lindel f Property with Respect to an Ideal REZA SALEKI R Saleki@math iut ac ir January 11 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 ir2010 MSC Primary 54D20 Secondary 54D30 Keywords Lindel f Weakly Lindel f Almost Lindel f Ideal Codense Ideal Continuous Function Open Map Closed Map Paracompact Lightcompact Locally FiniteAbstract We define being Lindel f with respect to an ideal and investigate basic properties of the concept itsrelation to known concepts and its preservation by functions subspaces pre images and products An ideal is a nonempty collection of subsets of X closed under operations of subset heredity andfinite union finite additivity If in addition the ideal is closed under the operation of countableunions it is called ideal We denote a topological space X with an ideal I defined on X as X I and call X I an ideal topological space A space X I is said to be I Lindel f or Lindel f with respect to I if every open cover U of Xhas a countable subcollection V such that X V I a space is Lindel f iff it is Lindel f Frolik defines a space to be weakly Lindel f if every opencover U of the space has a countable subcollection V such that X V We now show that weaklyLindel f spaces are a special case of Lindel f with respect to an ideal If X is a space we denotethe ideal of nowhere dense sets by N and the ideal of meager first category subsets by M An ideal I on X is said to be codense if I Let X be a space 1 X is weakly Lindel f iff X is N Lindel f 2 X is weakly Lindel f iff X is Lindel f with respect to some codense ideal 3 If X is a Baire space then X is weakly Lindel f iff X is M Lindel f A space X is said to be countably compact with respect to an ideal I or simply countably I compact if every countable open cover of the space admits a finite subcollection which covers allthe space except for a set in the ideal It is shown that a space X with an ideal I is countably I compact if and only if every locally finite collection of non ideal subsets is finite A space X with anideal I is said to be paracompact with respect to I or simply I paracompact if every open cover ofthe space admits a locally finite open refinement not necessarily a cover which covers all the spaceexcept for a set in the ideal Let X be a space with an ideal I such that X is I paracompactand I It is then shown that countable I compactness and I compactness are equivalent Special cases include countable compactness is equivalent to compactness in paracompact spaces lightcompactness is equivalent to quasi H closedness in almost paracompact spaces and countable meager compactness is equivalent to meager compactness in meager paracompact Baire spaces

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اوليه54D20و ثانويه54D30