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A Study of Some Aspects of Topologized Groups

Thesis Info

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Author

Noreen, Rafaqat

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/123456789/11975/1/Rafaqat%20Noreen_Maths_2019_Comsats_PRR.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676725417980

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A Study of Some Aspects of Topologized Groups A topological group is a mathematical object along with algebraic and topological structures. In this thesis our motivation is to study topological properties in the presence ofalgebraic structure particularly when the mappings are not continuous. It is a common question in topological algebra that how the relationship between topological properties depends upon the underlying algebraic structure. It is noticed that weaker the restriction in between algebraic structure and topology are, the larger is the class we obtain. Since we have weakened the condition of continuity and replaced it with the weaker form that is semi continuity and irresolute mapping and resultantly obtained s-(S-, Irresolute, Irr-) paratopologized or topologized groups. The purpose of this thesis is to give a mostly self contained study of s-(S-, Irresolute, Irr-) paratopologized or topologized groups. It is shown that every irresolute paratopological group is Irr-paratopological group as well as s-topological group. Every paratopological groups is s-paratopological group as well as S-paratopological group, and every Irr- paratopological group is S-paratopological group. Counter examples are given to show that such paratopological groups are generalized forms of the corresponding topologized groups. We have also defined and studied quasiboundedness of irresolute paratopological groups and s-paratopological groups. New notions- quasi bounded and - quasi bounded homomorphisms are introduced and discussed. In Chapter number six, we have defined semi-quotient mappings which are stronger than semi continuous mappings. Various interesting and important results on semi-quotients of paratopologized groups are proved. We have also studied semi connectedness for topologized groups.
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مولانا خلیل الرحمن

مولانا خلیل الرحمن
افسوس کہ مولانا خلیل الرحمن صاحب سابق ناظم ندوۃ العلماء نے ۴؍ فروری ۱۹۳۶؁ء کی شب کو اپنے وطن سہارنپور میں اس دار فانی کو الوداع کہا، مولانائے مرحوم مولانا احمد علی صاحب محدث سہارنپوری (محشی بخاری و تلمیذ مولانا شاہ محمد اسحق دہلوی) کے چشم و چراغ تھے، مولانا احمد علی مرحوم پچھلی صدی کے آخری دور میں ہندوستان کے ان باکمالوں میں تھے جن کی مسندِ درس سے علم دین کی شمع روشن تھی اور تشنگانِ علم اس سرچشمہ سے سیراب ہونے کے لئے سینکڑوں میل کی منزلیں پاپیادہ طے کرکے وہاں تک پہنچتے تھے، مولانا خلیل الرحمن نے علم کے اسی گہوارہ میں آنکھ کھولی اور اپنے والد ماجد کے دامنِ فیض میں تعلیم و تربیت پاکر فارغ التحصیل ہوئے۔
مرحوم ندوۃ العلماء کے دور اول کے محسنین میں سے تھے، مولانا محمد علی مونگیریؒ ناظم ندوۃ العلماء کی معیت میں اس ملی و علمی خدمت میں شریک ہوئے اور آخر تک رہے، مرحوم خوش خلق، متواضع، رحمدل، اور عزیزوں سے دلی محبت فرمانے والے تھے، اتفاق وقت کہ دارالعلوم ندوۃ العلماء کی ہنگامہ خیز اسٹرائک کا واقعہ انہی کے دور نظامت میں پیش آیا تھا، اس نازک وقت اور ناسازگار حالات میں بھی مولانائے مرحوم دارالعلوم کے طلبہ کے ساتھ جس عدیم المثال شفقت و محبت سے پیش آئے، اسکی یاد اس عہد کے فارغ التحصیل علمائے ندوہ کے دلوں میں ہمیشہ کے لئے باقی رہ گئی، اور مدت گزرنے کے بعد انہیں جب کبھی مرحوم سے شرف ملاقات کا موقع حاصل ہوا انھوں نے ان کے دل کو شفقت و محبت سے لبریز پایا ندوۃ العلماء کی خدمات انجام دینے کے علاوہ مرحوم کی زندگی کا ایک اہم کارنامہ صحیح بخاری کے اس نادر نسخہ کی اشاعت ہے، جس پر ان کے والد ماجد کے حواشی ثبت ہیں، یہ...

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