A function space triple of a compact polyhedron into an open set in Euclidean space

Sakai, Katsuro

doi:10.1090/S0002-9939-1990-0991709-3

A function space triple of a compact polyhedron into an open set in Euclidean space

Author: Katsuro Sakai
Journal: Proc. Amer. Math. Soc. 108 (1990), 547-555
MSC: Primary 57N20; Secondary 54C35, 58B05, 58D15
DOI: https://doi.org/10.1090/S0002-9939-1990-0991709-3
MathSciNet review: 991709
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Abstract: Let be a non-zero dimensional compact Euclidean polyhedron and an open set in Euclidean space ${{\mathbf{R}}^r}\left( {r > 0} \right)$ . The spaces of (continuous) maps, Lipschitz maps and PL maps from to are denoted by $C\left( {X,Y} \right)$ , ${\text{LIP}}\left( {X,Y} \right)$ and ${\text{PL}}\left( {X,Y} \right)$ , respectively. We prove that the triple

$\displaystyle \left( {C\left( {X,Y} \right),{\text{LIP}}\left( {X,Y} \right){\text{PL}}\left( {X,Y} \right)} \right)$

is an $\left( {s,\Sigma ,\sigma } \right)$ -manifold triple, where $s = {\left( { - 1,1} \right)^\omega }$ ,

$\displaystyle \Sigma = \left\{ {x \in s\vert\sup \left\vert {x\left( i \right)}... ...\,{\text{except}}\,{\text{for}}\,{\text{finitely}}\,{\text{many}}\,i} \right\}.$

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