Correspondence between interval \(\pi\)-equivalences and Sd-functions. (English) Zbl 0672.03035

Witzany, Jiří

Commentat. Math. Univ. Carol. 30, No. 1, 175-187 (1989).

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

Keywords:

revealed class; nearness; interval equivalences; alternative set theory