Properties and Counterexamples on Generalizations of Lindelof Spaces

In this thesis, generalizations of Lindelof spaces that depend on open covers and regularly open covers are studied. Namely: nearly Lindelof, almost Lindelof and weakly Lindelof spaces. And generalizations of regular-Lindelof spaces that depend on regular covers are also studied. Namely: nearly regularLindelof, almost regular-Lindelof and weakly regular-Lindelof spaces. Some properties and characterizations of these six generalizations of Lindeiof spaces are given. The relations among them are studied and some counterexamples are given in order to prove that the studied generatizations are proper generalizations of Lindelof spaces. Subspaces and subsets of these spaces are studied. We show that some subsets of these spaces inherit these covering properties and some others they do not Moreover, semiregular property on these spacess is studied to establish that all of these properties are semiregutar properties on the contrary of Lindelof property which is not a semiregular property. Mappings and generalized cont inuous functions are also studied on these generalizations and we prove that these properties are topological properties. Relations and some properties of many decompositions of continuity and generalized continuity that recently defined and studied are given. Counterexamples are also given to establish the relations among these generalizations of continuity. We show that some proper mappings preserve these topological properties such as: 6-continuity preserves nearly Lindelof property. O-continuity preserves almost Lindelof property. Rmaps preserve nearly regular-Lindelof property. Almost continuity preserves weakly Lindelof, almost regular-Lindelof and weakly regular-Lindelof properties. Moreover, we give some conditions on the functions or on the spaces to prove that weak forms of continuity preserve some of these covering properties under these conditions. The product property on these generalizations is also studied. We show that these topological properties, as in the case of most non-compact properties, are not preserved by product, even under a finite product. Some conditions are given on these generalizations to prove that these properties are preserved by finite product under these conditions. We show that, in weak P-spaces, finite product of nearly Lindelof spaces is nearly Lindelof and finite product of weakly Lindelof spaces is almost Lindelof.

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