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Definable continuous selections of set-valued maps in o-minimal expansions of the real field

Sokantika S., Thamrongthanyalak A.

Bulletin of the Polish Academy of Sciences. Mathematics

2017

Vol. 65, no. 2

97--105

Let T be a set-valued map from a subset of Rn to Rm. Suppose (R;+,⋅,T) is o-minimal. We prove that (1) if for every x ∈ Rn, each connected component of T(x) is convex, then T has a continuous selection if and only if T has a continuous selection definable in (R;+,⋅,T); (2) if n = 1 or m = 1, then T has a continuous selection if and only if T has a continuous selection definable in (R;+,⋅,T).