Mappings on weakly Lindelöf and weakly regular-Lindel¨of spaces

[EN] In this paper we study the effect of mappings and some decompositions of continuity on weakly Lindelöf spaces and weakly regular-Lindelöf spaces. We show that some mappings preserve these topological properties. We also show that the image of a weakly Lindelöf space (resp. weakly regular-Lindelöf space) under an almost continuous mapping is weakly Lindelöf (resp. weakly regular-Lindelöf). Moreover, the image of a weakly regular-Lindelöf space under a precontinuous and contracontinuousmapping is Lindelöf.Fawakhreh, AJ.; Kiliçman, A. (2011). Mappings on weakly Lindelöf and weakly regular-Lindel¨of spaces. Applied General Topology. 12(2):135-141. doi:10.4995/agt.2011.1647.SWORD13514112

Mappings and decompositions of continuity on almost Lindelöf spaces

A topological space X is said to be almost Lindelöf if for every open cover {Uα:α∈Δ} of X there exists a countable subset {αn:n∈ℕ}⊆Δ such that X=∪n∈ℕCl(Uαn). In this paper we study the effect of mappings and some decompositions of continuity on almost Lindelöf spaces. The main result is that a image of an almost Lindelöf space is almost Lindelöf

Generalizations of Lindelöf Properties in Bitopological Spaces

A bitopological space (X, τ 1, τ 2) is a set X together with two (arbitrary) topologies τ 1 and τ 2 defined on X. The first significant investigation into bitopological spaces was launched by J. C. Kelly in 1963. He recognized that by relaxing the symmetry condition on pseudo-metrics, two topologies were induced by the resulting quasipseudo- metrics. Furthermore, Kelly extended some of the standard results of separation axioms in a topological space to a bitopological space. Some such extension are pairwise regular, pairwise Hausdorff and pairwise normal spaces. There are several works dedicated to the investigation of bitopologies; most of them deal with the theory itself but very few with applications. In this thesis, we are concerned with the ideas of pairwise Lindelöfness, generalizations of pairwise Lindelöfness and generalizations of pairwise regular-Lindelöfness in bitopological spaces motivated by the known ideas of Lindelöfness, generalized Lindelöfness and generalized regular-Lindelöfness in topological spaces. There are four kinds of pairwise Lindelöf space namely Lindelöf, B-Lindelöf, s- Lindelöf and p-Lindelöf spaces that depend on open, i-open, τ 1 τ2 -open and p-open covers respectively introduced by Reilly in 1973, and Fora and Hdeib in 1983. For instance, a bitopological space X is said to be p-Lindelöf if every p-open cover of X has a countable subcover. There are three kinds of generalized pairwise Lindelöf space namely pairwise nearly Lindelöf, pairwise almost Lindelöf and pairwise weakly Lindelöf spaces that depend on open covers and pairwise regular open covers. Another idea is to generalize pairwise regular-Lindelöfness to bitopological spaces. This leads to the classes of pairwise nearly regular-Lindelöf, pairwise almost regular-Lindelöf and pairwise weakly regular-Lindelöf spaces that depend on pairwise regular covers. Some characterizations of these generalized Lindelöf bitopological spaces are given. The relations among them are studied and some counterexamples are given in order to prove that the generalizations studied are proper generalizations of Lindelöf bitopological spaces. Subspaces and subsets of these spaces are also studied, and some of their characterizations investigated. We show that some subsets of these spaces inherit these generalized pairwise covering properties and some others, do not. Mappings and generalized pairwise continuity are also studied in relation to these generalized pairwise covering properties and we prove that these properties are bitopological properties. Some decompositions of pairwise continuity are defined and their properties are studied. Several counterexamples are also given to establish the relations among these generalized pairwise continuities. The effect of mappings, some decompositions of pairwise continuity and some generalized pairwise openness mappings on these generalized pairwise covering properties are investigated. We show that some proper mappings preserve these pairwise covering properties such as: pairwise δ-continuity preserves the pairwise nearly Lindelöf property; pairwise θ- continuity preserves the pairwise almost Lindelöf property; pairwise almost continuity preserves the pairwise weakly Lindelöf, pairwise almost regular-Lindelöf and pairwise weakly regular-Lindelöf properties; and pairwise R-maps preserve the pairwise nearly regular-Lindelöf property. Moreover, we give some conditions on the maps or on the spaces which ensure that weak forms of pairwise continuity preserve some of these generalized pairwise covering properties. Furthermore, it is shown that all the generalized pairwise covering properties are satisfy the pairwise semiregular invariant properties where some of them satisfy the pairwise semiregular properties. On the other hand, none of the pairwise Lindelöf properties are pairwise semiregular properties. The productivity of these generalized pairwise covering properties are also studied. It is well known by Tychonoff Product Theorem that compactness and pairwise compactness are preserved under products. We show by means of counterexamples that in general the pairwise Lindelöf, pairwise nearly Lindelöf and similar properties are not even preserved under finite products. We give some necessary conditions, for example the P-space property; under which these generalized pairwise covering properties become finitely productive

Tree-Structured and Direct Parametric Regression Models for the Subdistribution of Competing Risks

Traditionally, the regression analysis for competing risks survival time is based on the cause-specific hazard that treat failures from causes other than the cause of interest as censored observations. That includes technique such as the Cox proportional hazard model. The modelling of hazard rate may or may not match the objective of investigator. It is often more desirable to investigate the subdistribution function, because cause-specific hazard doesn’t obviously give the information about proportion of individuals experiencing a cause of interest. Furthermore, the subdistribution and cause-specific hazard function are not interchangeable. Thus, if we intended to draw inference from subdistribution function, then we must model on subdistribution function directly or indirectly. Sometimes, we do not only intend to investigate the relationship between response and covariates through regression analysis, but also we want to identify the presence of subgroup of individuals in our data. We could then utilize tree-structured regression for this purpose. In this thesis, we developed statistical methods for competing risks data analysis through direct, indirect and parametric subdistribution modelling. Indirect model is employed via hazard of subdistribution. Evaluation of the performance of proposed methods is conducted through series of simulation studies as well as real data application. We developed four methods: 1) a method to categorize continuous covariate by considering the competing risks survival time outcome variables, called outcome-oriented categorization method, 2) a tree-structured competing risks regression to extract meaningful sub-groups of subjects determined by the value of covariates, 3) a hybrid model which boost the available subdistribution hazards regression by ugmenting it with tree-structured regression resulted from the previous step, 4) two kinds of parametric direct subdistribution model. These models are constructed based on non-mixture cure model. The first model is developed by taking into account the fraction of individuals who did not experience the event of interest in the long term. The second model is developed by reparameterizing the first model in order to mimic Gompertz distribution which allows no immune fraction. Research finding is as follows: 1) Method of outcome-oriented categorization based on deviance statistic is the best. The application of the method to contraceptive discontinuation data showed good result. 2) Regression tree for competing risks data can uncover the structure of data and yield the sub-group of individuals with a clear description based on their covariates. The application of the method to contraceptive discontinuation data showed good result. Extensive Monte Carlo simulation suggests the method has good performance in identifying the structure of data. 3) Application of the hybrid model to the contraceptive discontinuation data showed that the hybrid model is better than the available subdistribution regression in terms of AIC. 4) By using some well known kernel distribution, the parametric direct subdistribution models are developed. The maximum likelihood estimations are carried out simultaneously for all causes of event. In Bone Marrow Transplantation (BMT) data analysis, the first proposed model gave noticeably good fit to the nonparametric counterpart. The second proposed model is fitted to contraceptive discontinuation data and showed that Gompertz-like subdistribution with Gompertz kernel is the best fit

nv-Lindelöfness

Our work aims to study nearly ν-Lindelöf (briefly. nν-Lindelöf) space in generalized topological spaces. Moreover, some mappings and decompositions of continuity are studied. The main result that we obtained on is the effect of (δ,δ’)-continuous function on nν-Lindelöf space

Pairwise Semiregular Properties on Generalized Pairwise Lindelöf Spaces

Let (X, τ1, τ2) be a bitopological space and (X, τs(1,2), τs(2,1)) its pairwise semiregularization. Then a bitopological property P is called pairwise semiregular provided that (X, τ1, τ2) has the property P if and only if (X, τs(1,2), τs(2,1)) has the same property. In this work we study pairwise semiregular property of (i, j)-nearly Lindelöf, pairwise nearly Lindelöf, (i, j)-almost Lindelöf, pairwise almost Lindelöf, (i, j)-weakly Lindelöf and pairwise weakly Lindelöf spaces. We prove that (i, j)-almost Lindelöf, pairwise almost Lindelöf, (i, j)-weakly Lindelöf and pairwise weakly Lindelöf are pairwise semiregular properties, on the contrary of each type of pairwise Lindelöf space which are not pairwise semiregular properties

Topological Groups: Yesterday, Today, Tomorrow

In 1900, David Hilbert asked whether each locally euclidean topological group admits a Lie group structure. This was the fifth of his famous 23 questions which foreshadowed much of the mathematical creativity of the twentieth century. It required half a century of effort by several generations of eminent mathematicians until it was settled in the affirmative. These efforts resulted over time in the Peter-Weyl Theorem, the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups, and finally the solution of Hilbert 5 and the structure theory of locally compact groups, through the combined work of Andrew Gleason, Kenkichi Iwasawa, Deane Montgomery, and Leon Zippin. For a presentation of Hilbert 5 see the 2014 book “Hilbert’s Fifth Problem and Related Topics” by the winner of a 2006 Fields Medal and 2014 Breakthrough Prize in Mathematics, Terence Tao. It is not possible to describe briefly the richness of the topological group theory and the many directions taken since Hilbert 5. The 900 page reference book in 2013 “The Structure of Compact Groups” by Karl H. Hofmann and Sidney A. Morris, deals with one aspect of compact group theory. There are several books on profinite groups including those written by John S. Wilson (1998) and by Luis Ribes and ‎Pavel Zalesskii (2012). The 2007 book “The Lie Theory of Connected Pro-Lie Groups” by Karl Hofmann and Sidney A. Morris, demonstrates how powerful Lie Theory is in exposing the structure of infinite-dimensional Lie groups. The study of free topological groups initiated by A.A. Markov, M.I. Graev and S. Kakutani, has resulted in a wealth of interesting results, in particular those of A.V. Arkhangelʹskiĭ and many of his former students who developed this topic and its relations with topology. The book “Topological Groups and Related Structures” by Alexander Arkhangelʹskii and Mikhail Tkachenko has a diverse content including much material on free topological groups. Compactness conditions in topological groups, especially pseudocompactness as exemplified in the many papers of W.W. Comfort, has been another direction which has proved very fruitful to the present day

Allied subsets of topological groups and linear spaces

1, Basic theory, 4 || 2. Allied sets in linear spaces, 17 || 3. Cauchy nets and completeness, 29 || 4, Subgroups and subspaces. 39 || 5. Locally compact subsets of topological linear spaces. 46 || 6. Allied sets with respect to different topologies, 54 || 7. Decomposition theorems. 62 || 8. Open decomposition. 72 || 9. Applications to lattices, 83 || 10. Allied families, 91 || Appendix 1. Nets in A + pos b , 96 || Appendix 2. The Mackey topology of a subspace. 99 || Appendix 3. Publication of results, 100 || References 10