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On continuous approximations. (English) Zbl 0888.54024

Császár, Á. (ed.), Topology with applications. Proceedings of the 7th colloquium, Szekszárd, Hungary, August 23–27, 1993. Budapest: János Bolyai Mathematical Society. Bolyai Soc. Math. Stud. 4, 419-425 (1995).
Multifunctions are investigated the values of which being graphs of polynomials. If \(X\) is a normed space and \(V\subset X\) a convex dense subset of \(X\) then, as is proved, for every lsc multifunction \(F:P\multimap X\) defined on the paracompactum \(P\) and with convex, not necessarily closed values and for every non-decreasing function \(g:(0,+\infty)\to (0,+\infty)\) there exists a continuous mapping \(f:P\times (0,+\infty)\to V\) such that \(\text{dist} (f(p,\varepsilon),F(p))< g(\varepsilon)\) for any \((p,\varepsilon)\in P\times (0,+\infty)\). Other conditions equivalent to the above are formulated and it is shown by an example that there exists a Hausdorff continuous multifunction \(F:[0,1]\multimap \mathbb{R}^2\) such that all values of \(F\) are graphs of polynomials and \(F\) has no continuous selectors.
For the entire collection see [Zbl 0871.00038].

MSC:

54C65 Selections in general topology
54C60 Set-valued maps in general topology
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