Mary Ellen Rudin

Mary Ellen Rudin
Umumiy maʼlumotlar
Tavalludi	7-dekabr 1924 Hillsboro
Vafoti	18-mart 2013 Madison
Qardosh loyihalar
Vikipediyadagi maqola Vikimaʼlumotlar sahifasi

Mary Ellen Rudin (qizlik familiyasi Estill, 1924-yil 7-dekabr — 2013-yil 18-mart) — toʻplamlar nazariyasi topologiyasi sohasida ixtisoslashgan amerikalik matematik. U bir qator muhim yutuqlari bilan tanilgan boʻlib, ularning orasida Dauker fazosining ilk bor yaratilishi va Nikiel farazining birinchi marta isbotlanishi alohida ahamiyatga ega.

Iqtiboslar

Suslin gipotezasi sodda eshitiladi. „Sanoqli“ va „sanoqsiz“ tushunchalarining maʼnosini tushunadigan har bir kishi bu ustida „ishlashi“ mumkin. Aslida esa, u juda ayyorona (murakkab). Bunda odam quradigan standart andozalar mavjud. Shuningdek, odam yoʻl qoʻyadigan standart xato mulohazalar ham bor. Va deyarli har bir kishi ushbu muammo bilan shugʻullanganida duch keladigan standart „unchalik-aks-misol-emas“ holatlar mavjud. Stanley Tennenbaum va boshqalar Suslin gipotezasi Zermelo-Fraenkel toʻplamlar nazariyasi aksiomalari bilan mos ravishda ham rost, ham yolgʻon boʻlishi mumkinligini koʻrsatib berishdi.

Souslin's conjecture sounds simple. Anyone who understands the meaning of countable and uncountable can "work" on it. It is in fact very tricky. There are standard patterns one builds. There are standard errors in judgement one makes. And there are standard not-quite-counter-examples which almost everyone who looks at the problem happens upon. S. Tennenbaum and others have shown that that it is consistent with the axioms of Zermelo-Fraenkel set theory that Souslin's conjecture be either true or false.

Ushbu maqolaning maqsadi (tanlov aksiomasidan tashqari hech qanday toʻplamlar nazariyasi shartlaridan foydalanmasdan) shunday normal Hausforff $X$ boʻshligʻini qurishdan iboratki, uning [0, 1] yopiq birlik kesmasi $I$ bilan Dekart koʻpaytmasi normal emasdir. Bunday boʻshliq koʻpincha Dowker boʻshligʻi deb ataladi. Bunday boʻshliqning mavjudligi haqidagi savol qadimiy va tabiiy savollardan biridir...

The purpose of this paper is to construct (without using any set theoretic conditions beyond the axiom of choice) a normal Hausforff space X whose Cartesian product with the closed unit interval I is not normal. Such a space is often called a Dowker space. The question of the existence of such a space is an old and natural one ...

Geometrik topologiya 1950-yillarda, differensial topologiya esa 1960-yillarda haqiqatda hukmron yangi topologik mavzular boʻldi. Algebratik topologiya esa bu ikki jarayonning hech birida orqa qatorda qolib ketmadi. Ammo 1960-yillarda biz shugʻullanayotgan topologiya sohasiga chuqur ta’sir koʻrsatgan bir voqea sodir boʻldi. ...Paul Cohen toʻplamlar nazariyasining odatiy aksiomalari doirasida kontinuum gipotezasining (CH) yolgʻon boʻlishi mumkinligini isbotlab berdi. Oʻz-oʻzidan bu teorema topologiya uchun unchalik koʻp oqibatlarga ega emas, chunki faqat „not-CH“ (kontinuum gipotezasining inkor etilishi) ning oʻzi bilan juda kam narsa qilish mumkin. Biroq, „forcing“ deb ataladigan isbotlash texnikasining Bul algebrasi, qisman tartib va ajoyib kombinatorik ifodalar tiliga tarjima qilinishi mavhum boʻshliqlar bilan bogʻliq boʻlgan turli xil topologik muammolarga tatbiq etilishi mumkin.

Geometric topology was really the dominant new topological theme in the 1950's and differential topology in the 1960's. Algebraic topology did not take a back seat in either development. But something happened in the 1960's which had profound effect upon the part of topology we are concerned with. ... Paul Cohen proved that it is consistent with the usual axioms for set theory that the continuum hypothesis be false. In itself this theorem has few consequence in topology for there is very little one can do with not-CH alone. But the technique of proof, called forcing, has translations into Boolean algebra terms, into partial order terms, into terms which lead to remarkable combinatorial statements which are applicable to a wide variety of topological problems related to abstract spaces.

Mary Ellen Rudin

	Bizning ilk yuzma-yuz uchrashuvimiz 1970-yilning yozida Nitssadagi IMU Kongressida boʻlib oʻtdi. Doʻstim va hamkasbim András Hajnal bilan biz u bilan koʻrishishga mushtoq edik va bu u Nitssaga kelishi bilanoq sodir boʻldi. Uning bizga aytgan birinchi jumlasi: „Men hozirgina Dowker boʻshligʻi mavjudligini isbotladim“, — boʻldi; yaʼni, birlik kesma bilan koʻpaytmasi normal boʻlmagan normal fazoni topganini aytdi. Ushbu jumlaning naqadar vazmin ekanini his qilish uchun shuni bilish kerakki, bu u 1960-yillardagi umumiy topologiyaning eng muhim ochiq muammosini hal qilganini anglatar edi.
	Our first meeting in person took place at the IMU Congress in Nice in the summer of 1970. Together with my friend and collaborator András Hajnal we were eager to meet her, and this happened right after she arrived in Nice. Her first sentence to us was “I just proved that there is a Dowker space;” i.e., a normal space whose product with the unit interval is not normal. To appreciate the weight of this sentence, one should know that this meant she solved the most important open problem of general topology of the 1960s.
	— István Juhász