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Selection Principles and Weaker Forms of Menger, Rothberger and Hurewicz Properties in Topology

Thesis Info

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Author

Sabah, Amani

Program

PhD

Institute

COMSATS University Islamabad

City

Islamabad

Province

Islamabad.

Country

Pakistan

Thesis Completing Year

2019

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/10626/1/Amani%20Sabah_Maths_2019_Comsats_PRR.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727157414

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In this research work, our focus is to study certain covering properties in topological spaces by using semiopen covers and in bitopological spaces by using pairwise open covers and their generalizations. This work deals with Menger type, Rothberger type and Hurewicz type covering properties. The notions of semiMenger (semiRothberger), almost semiMenger (almost semiRothberger), star semiMenger (star semiRothberger), almost star semiMenger (almost star semiRothberger) and strongly star semiMenger (strongly star semiRothberger) are defined and corresponding covering properties are investigated. Furthermore, the concepts of semiHurewicz, almost semiHurewicz and star semiHurewicz spaces are introduced and their properties are studied. Relations between the newly defined notions and classical selection principles are established. Similarities and differences among each other are discussed. We have also studied the prototypes of some selection principles using p-open covers in bitopological spaces.
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آزادی ایک نعمت ہے

آزادی ایک نعمت عظیم ہے
صبحِ آزادی کا سورج جسم و جاں پہ قرض ہے
اس کی کِرنوں کی حفاظت اب تمھارا فرض ہے
آزادی خدا کی بہت بڑی نعمت ہے اور اس نعمت کا شکر ادا کرنا بہت بڑی سعادت ہے۔ شکر در اصل نعمت کی فراوانی کا سبب ہوا کرتا ہے اور کفرانِ نعمت زوال کا باعث بنتا ہے۔ شاکرین کے در پرنعمتیں مسلسل دستک دیتی رہتی ہیں۔
آج ہم اپنے گریبان میں منہ ڈال کر جھانکیں کہ ہم نے آزادی کی نعمت کا کس حد تک شکر ادا کیا ہے اور خدا اورمخلوقِ خدا سے جو وعدے کیے تھے انہیں کہاں تک پورا کیا ہے؟
ہم نے نعرہ لگایا تھا کہ’’ ہم پاکستان میں قرآن کا قانون جاری کریں گے۔ نبی کریم صلی اللہ علیہ و آلہٖ وسلم کی سنت کی روشنی میں زندگی کا سفر طے کریں گے۔ نیکی کا عَلم بلند کریں گے اور بدی کو سرنگوں کر کے چھوڑیں گے ، گناہوں کی اندھیری رات میں نیکی کے دیئے جلائیں گے اور شبِ تیرہ کی تیرگی کا جنازہ نکال کر دم لیں گے۔ جبر و استبدادکی کالی گھٹائیں چھٹ جائیںگی۔ عدل وانصاف کا قاضی محمودو ایاز کو ایک صف میں کھڑا کر دے گا۔ غربت کے مہیب سائے رخصت ہو جائیں گے۔ تنگدستی او محتاجی کے عفریت کو منہ کی کھانی پڑے گی۔ خوشحالی کا آفتاب طلوع ہوگا اور افلاس زدہ انسان امن و عافیت کے گہوارے میں خوشی کے نغمے گاتے ہوئے زندگی بسر کریں گے۔
یہ تھے وہ مقاصد جن کے حصول کے لیے برصغیر کے لاکھوں بلکہ کروڑوں انسانوں نے ناقابلِ فراموش قربانیاں دیں۔ اسی ارض پاک کے لیے ہمیں آگ اور خون کے دریاؤں سے گزرنا پڑا۔ یہ پاکستان نہ تو حسینوں کی اداؤںسے بنا اور نہ ہی یہ مغنّیہ کی صداؤں سے بنا اور نہ ہی...

Digital Marketing and Creative Economy Theory in Economic Development During the Pandemic

Changes in the world require a company to make innovations that are necessary in order to survive the onslaught of other companies' innovations, especially similar companies. A new economic concept that focuses on information and creativity that relies on creative ideas and knowledge from human resources for the main production factors. Creative economynowadays it is increasingly being carried out by the people, especially the younger generation in country because they feel this is a profession that is suitable to be done. Information technology is also needed in the distribution, promotion and sales transaction processes so that the process runs more effectively and efficiently. The potential for the development of this industry is due to several factors, namely the development of information technology as the main factor develops rapidly, access to information centers via the internet is much easier, the social innovation process runs smoothly, each region has a unique local cultural potential and the openness of society to modern culture, there are sources of knowledge such as many campuses that are a source of quality human resources, have high creativity and innovation.

Primary Antiretroviral Drug Resistance Profiles Among Hiv-1 Therapy Naïve Patients in Two Referral Hospitals in Kenya

Objective: To characterize antiretroviral drug resistance mutations among drug naïve patients in two referral hospitals in Kenya Background: Acquired immunodeficiency syndrome (AIDS) caused by Human immunodeficiency virus (HIV) was first described in 1982. Since then the virus has spread globally to infect millions of people. HIV was first described in Kenya in the period between 1984/1985. Currently, Kenya has an estimated HIV-1 prevalence of 6.2% with a country population of about 40 million people. With the introduction of antiretroviral drugs, the survival of most HIV patients has been prolonged markedly. However this is greatly threatened by increasing rates of antiretroviral dug resistance, which may eventually lead to suboptimal treatment outcomes. Methods: The aim of this study was to determine antiretroviral mutation profiles among drug naïve patients in two referral hospitals in Kenya. Antiretroviral naïve HIV patients in Aga Khan University Hospital and Thika Level 5 Hospital were consecutively recruited to participate in the study. Participants with viral loads >1000 copies/millilitre had their samples screened for antiretroviral resistance mutations by genotypic testing. A total of 121 participants were recruited into this study from two centres. Eighty four participants had their samples successfully genotyped for drug resistance mutations. Results: A total of five NRTI mutations (two Y115F, K219Q, K219E, and T215F) and one V106I mutation against NNRTIs were found among participants in this study. One study participant had one protease mutation, M46L. The estimated primary antiretroviral resistance rate against reverse transcriptase inhibitors was 8.7 % (95% CI 4.0-17.7). Resistance against protease inhibitors was found to be low at 1.46% (95% CI 0.26-7.9). The most common viral subtype was A1 at 52%. Others subtypes included D at 17%, subtype C at 13%, viral recombinants (CRF01_AE, CRF02_AG) at 12%, B at 3% and others subtypes (H, J) at 3%. Conclusion: Antiretroviral drug resistance mutations are showing an increasing trend among therapy naïve patients since the introduction of antiretroviral therapy in Kenya in the early 2000. This might affect the efficacy of antiretroviral regimens used for treating HIV patients. Funding: The study was co-funded by the Aga Khan University Research Council Grant (URC Grant Project 102001KEN Under Dr Nancy Okinda) and Aga Khan University Postgraduate Medical Education Seed Funding.