Generalizations of paralindelӧf property in bitopological spaces
The idea of bitopological spaces was initiated first by Kelly in 1969 which de- fined as two topologies defined on one set (Kelly, 1963). Furthermore, Kelly has extended several topological properties and their results to a bitopological space such as pairwise Hausdorff, pairwise regular, pairwise n...
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| Format: | Thesis |
| Language: | English |
| Published: |
2015
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/65437/1/FS%202015%2039IR.pdf |
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| Summary: | The idea of bitopological spaces was initiated first by Kelly in 1969 which de- fined as two topologies defined on one set (Kelly, 1963). Furthermore, Kelly has extended several topological properties and their results to a bitopological space such as pairwise Hausdorff, pairwise regular, pairwise normal spaces. After that, many authors deal with the extensions to bitopology theory and also with its applications. The main goal of this thesis is to study and focus on the notions of pairwise paralindel¨ofness, pairwise para-m-lindel¨of and generalizations of pairwise paralindel¨ofness in bitopological spaces derived by the well-known notions of paralindel¨of, para-m-lindel¨of and generalized paralindel¨f in topological settings. During this work, we define five types of paralindel¨of spaces in bitopological setting. Namely, paralindel¨of, FHP-pairwise paralindel¨of, RR-pairwise paralindel¨of, p-paralindel¨of and pairwise para-m-lindel¨of spaces. In the spaces of paralindel¨of and p-paralindel¨of, they are depended on open and τ1τ2-open covers respectively. Whereas the FHP-pairwise paralindel¨of and RR-pairwise paralindel¨of spaces are both depended on i-open cover. The difference between these spaces is on the definition of the refinement of their covers. For example, a bitopological space X is called p-paralindel¨of if every τ1τ2-open cover of X admits τ1τ2-open refinement p-locally countable. The generalizations of pairwise paralindel¨of space are pairwise nearly paralindel¨of and pairwise almost paralindel¨of spaces that are defined by open covers and pairwise regular open covers. In this study we state some characterizations of pairwise paralindel¨of space and its generalizations. Further, the relationships between these concepts are studied and supported by examples and even counterexamples. In addition, their subspaces and subsets are introduced, and investigated some characterizations and their behavior. Moreover, we present the idea of pairwise lo cally generalizations of paralindel¨ofness in bitopological spaces. For example, pairwise locally nearly paralindel¨of and pairwise locally almost paralindel¨of spaces. Further, we introduce the notion of one-point extension of pairwise almost paralindel¨of spaces and study some properties of this notion. This research is also concerned on mappings and generalized pairwise continuity and some relations among them were presented. The effect of some kinds of some generalized pairwise continuous and generalized pairwise open mappings on those generalized pairwise paralindel¨ofness are introduced and studied. For instance, pairwise perfect mappings with Lindel¨of inverse point (denoted by pairwise L-perfect mappings) map RR-pairwise paralindel¨of space to paralindel¨of bitopological space. Moreover, pairwise L-perfect and pairwise almost closed mappings are preserved pairwise almost paralindel¨of property. In contrast, we show that FHP-pairwise paralindel¨of space is not preserved under closed mappings which was given as an example. Furthermore, we add some extra conditions on the maps and sometimes on the spaces to guarantee as some weak types of pairwise mappings preserve the characters of some these generalized classes of pairwise covering properties. The product property of pairwise paralindel¨ofness and their generalizations are also investigated. We show that these covering properties are not preserved by product, even with the case of the finite product. To ensure the product is preserved, we restrict some conditions to these bitopological properties to prove that these properties are preserved by finite product under these conditions. For instance, in P-spaces, the product of two paralindel¨of bitopological spaces is also paralindel¨of bitopological space. As the theory of multifunctions arise, many of their classes are extended to bitopological settings. We extend some of multifunction to bitopological spaces as pairwise almost continuous, pairwise super continuous, pairwise δ- continuous and pairwise α-continuous multifunctions. Furthermore, we give some characterization of these concepts and study some their properties. In addition, we show the effectiveness of those multifunctions on the generalized pairwise paralindel¨of properties. |
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