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Proximity Approach to Problems in Topology and Analysis von Naimpally, Somashekhar (eBook)

  • Erscheinungsdatum: 01.10.2010
  • Verlag: De Gruyter Oldenbourg
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Proximity Approach to Problems in Topology and Analysis

Dieses Buch konzentriert das aktuelle Gesamtwissen zum Proximity-Konzept und stellt es dem Leser in gut strukturierter Form dar. Hauptaugenmerk liegt auf den vielfältigen Möglichkeiten, die sich aus dem Proximity-Konzept der räumlichen Nähe und seiner Verallgemeinerung im Nearness-Konzept ergeben.


    Format: PDF
    Kopierschutz: watermark
    Seitenzahl: 220
    Erscheinungsdatum: 01.10.2010
    Sprache: Englisch
    ISBN: 9783486598605
    Verlag: De Gruyter Oldenbourg
    Größe: 2119kBytes
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Proximity Approach to Problems in Topology and Analysis

" Intuitive Introduction (p. 1-2)

""The significant problems we face cannot be solved by the same level of thinking that created them."" - Albert Einstein

In this section, we propose to give a brief intuitive introduction to topology, proximity, and nearness. It shows that one can explain the concept of continuity and its various modifications even to persons who are not mathematicians. Teachers of calculus know how difficult it is to teach continuity in the classroom, see for example, Devlin's article on the website of the Mathematical Association of America:

Will the real continuous function please stand up? (May 2000)

It is not widely known that abstract pure mathematics has applications in daily life. There are many examples in mathematics but here we wish to explain topology and its applications. We use simple examples from day-to-day life to illustrate the concepts. Topology deals with the concept of nearness at various levels. Riesz first formulated this approach in an address to the International Congress of Mathematicians in Rome (1908).

Level 1: Consider a typical family {mother, father, son, daughter}. We say that a person is near the family if that person is blood related to some member of the family. Of course, every member of the family is near the family and the family must first exist to talk about nearness! Grandparents, aunts, uncles, cousins... are persons near the family though they are not in the family. There are many other ways of defining such nearness relations, e.g. one may say that a person is near the family if the person helps the family in some way. In this definition the family physician, the plumber, the mailman,... are near the family. This concept is axiomatized with a few simple obvious conditions and one gets the abstract concept of a topological space. This was accomplished by the Polish mathematician K. Kuratowski in 1922.

With a topological space, is associated another concept that of a continuous transformation. Suppose five years back P was a family physician of the family F. That is the Physician P was near the family F under the rule that P helps the family in some way. Today, after five years, both the Physician P and the family F have changed. If the Physician P is still near the family F, we say that it is a continuous relationship in day-to-day life and the same is said in mathematics. If for any reason, the Physician P ceases to be the family doctor of F, we say that the relationship is discontinuous. Thus a continuous transformation is one in which the nearness of a point to a set remains unchanged under that transformation. To recapitulate, in Topology, we have nearness relations between points and sets, together with continuous transformations which preserve these nearness relations.

Level 2: At this level we talk about nearness between two families, technically called a proximity. This idea, already present in Riesz's work, was thoroughly studied by the Soviet mathematician V. Efremovic around 1940 and published in 1951."

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