A Quantitative Multivalued Selection Theorem for Encoding Spaces of Continuum Cardinality

02/10/2020

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by   Donghyun Lim, et al.

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Previous work (Pauly Ziegler'13) had introduced and justified a qualitative and quantitative notion of uniform continuity for multi(valued)functions. This notion indeed passes several 'sanity' checks, including closure under restriction and composition and inducing compact images. We then show that every uniformly continuous pointwise compact multifunction from/to closed subsets of Cantor space (but not on the real unit interval) admits a single-valued selection with same modulus of continuity. This implies a quantitative strengthening of the qualitative Kreitz-Weihrauch MAIN Theorem (1985) about encoding spaces of continuum cardinality.