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n Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie - Elementêre topologie en berekeningsuniversaliteit - : navorsings- en oorsigartikel

Volume 27, Issue 4
  • ISSN : 0254-3486
  • E-ISSN: 2222-4173

Abstract

This paper attempts to define a general framework for computability on an arbitrary topological space X. The elements of X are taken as primitives in this approach - also for the coding of functions - and, except when X = N, the natural numbers are not used directly. The natural numbers, incidentally, are let in through the back door, as it were, only through the use of the concept of . For, in topology one uses the fact that the unions indexed by a natural number (seen as a finite ordinal) of open sets should be open. The topology determines the notion of and if we require that the computable functions be continuous then the role of the natural numbers becomes, once again, impossible to overlook. The conditions for being a that are given here, are necessary but far from sufficient for most purposes. This is done on purpose, so as to keep the discussion as general as possible. The usual notion of requires that a computational structure should have the

In hierdie bydrae word gepoog om 'n algemene raamwerk te beskryf waarin berekenbaarheid vir 'n willekeurige topologiese ruimte X gedefinieer kan word. Die elemente van X word in dié benadering beskou as die primitiewe voorwerpe - ook vir die kodering van funksies. Ons begin deur nodige (maar nie noodwendig voldoende) voorwaardes te stel waaraan 'n stel funksies wat as beskou word, moet voldoen. Die voorwaardes in die definisie van 'n wat gebruik word is ook nodig maar nie voldoende nie. 'n vir topologiese ruimtes word gedefinieer. Baie topologiese ruimtes waaroor ons graag sou bereken het wel die speldprikeienskap. Die hoofresultaat is dat ruimtes met die speldprikeienskap nie die , wat nodig is vir 'n bruikbare berekenbaarheidsbegrip, het nie. Die afwesigheid van die paarkoderingseienskap beteken dat ons nie op die gebruiklike wyse in dié topologiese berekeningsruimtes'n universele funksie kan definieer nie.

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/content/aknat/27/4/EJC20437
2008-12-01
2019-12-11

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