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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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 $ X$ be a non-zero dimensional compact Euclidean polyhedron and $ Y$ an open set in Euclidean space $ {{\mathbf{R}}^r}\left( {r > 0} \right)$. The spaces of (continuous) maps, Lipschitz maps and PL maps from $ X$ to $ Y$ 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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-0991709-3
Keywords: Space of (continuous) maps, space of Lipschitz maps, space of PL maps, polyhedron, convex set, $ \left( {s,\Sigma ,\sigma } \right)$-manifold triple
Article copyright: © Copyright 1990 American Mathematical Society

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