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Correspondence between interval \(\pi\)-equivalences and Sd-functions. (English) Zbl 0672.03035
If we want to investigate, e.g., the function \(s(x)=\sin (1/x)\) near 0 by nonstandard means, then it is suitable to use for it the topology described by the nearness \(x=_ sy=\{<x,y>\); (\(\forall z\), \(x\leq z\leq y)(s(x)\doteq s(z)\doteq s(y))\}\). This nearness has the property of convexity \(x<z<y \& x=_ sy\Rightarrow x=_ sz=_ sy\). Equivalences having this property are called interval equivalences. In topology in alternative set theory it is highly suitable to study also symmetries (transitivity is not demanded). Thus the author investigates interval symmetries generally. The main result of the paper is the assertion that every interval \(\pi\)-equivalence can be described in the above-mentioned way, i.e. that for every interval \(\pi\)-equivalence \(=^{+}\) there is an \(Sd^*_ V\) function F such that \(=^{+}=\quad =_ F.\)
Reviewer: K.Čuda
MSC:
03E70 Nonclassical and second-order set theories
54C30 Real-valued functions in general topology
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