A $Z$-set in a metric space $X$ is a closed subset $K$ of $X$ such that each
map of the Hilbert cube $Q$ into $X$ can uniformly be approximated by maps of
$Q$ into $X \setminus K$. The aim of the paper is to show that there exists a
functor of extension of maps between $Z$-sets of $Q$ [or $l_2$] to maps acting
on the whole space $Q$ [resp. $l_2$]. Special properties of the functor are
proved.Comment: 9 page

Cz Bessaga

H Toruńczyk

K Kuratowski

M Takesaki

N Aronszajn

O-H Keller

P Niemiec

PR Halmos

RD Anderson

TA Chapman

TO Banakh

VL Klee Jr.

Central European Journal of Mathematics

English

arXiv

AbstractIt is shown that if Ω = Q or Ω = ℓ 2, then there exists a functor of extension of maps between Z-sets in Ω to mappings of Ω into itself. This functor transforms homeomorphisms into homeomorphisms, thus giving a functorial setting to a well-known theorem of Anderson [Anderson R.D., On topological infinite deficiency, Michigan Math. J., 1967, 14, 365–383]. It also preserves convergence of sequences of mappings, both pointwise and uniform on compact sets, and supremum distances as well as uniform continuity, Lipschitz property, nonexpansiveness of maps in appropriate metrics.</jats:p

Piotr Niemiec

Crossref

Open Mathematics

Functor of extension in Hilbert cube and Hilbert space

Niemiec, Piotr

Jagiellonian Univeristy Repository

Niemiec Piotr

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https://ruj.uj.edu.pl/xmlui/bitstream/handle/item/3881/niemiec_functor_of_extension_2014.pdf?sequence=1&isAllowed=y