Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 }$,

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

.

References [Enhancements On Off] (What's this?)

  • [An] R. D. Anderson, Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 72 (1966), 515-519. MR 0190888 (32:8298)
  • [Ch] T. A. Chapman, Dense sigma-compact subsets of infinite-dimensional manifolds, Trans. Amer. Math. Soc. 154 (1971), 399-426. MR 0283828 (44:1058)
  • [Do] T. Dobrowolski, The compact $ Z$-set property in convex sets, Topology Appl. 23 (1986), 163-172. MR 855456 (87m:57016)
  • [Ge$ _{1}$] R. Geoghegan, On spaces of homeomorphisms, embeddings, and functions I, Topology 11 (1972), 159-177. MR 0295281 (45:4349)
  • [Ge$ _{2}$] -, On spaces of homeomorphisms, embeddings, and functions, II: The piecewise linear case, Proc. London Math. Soc. (3) 27 (1973), 463-483. MR 0328969 (48:7311)
  • [LV] J. Luukkainen and J. Väisälä, Elements of Lipschitz topology, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 3 (1977), 85-122. MR 515647 (80b:57015)
  • [Sa$ _{1}$] K. Sakai, The space of cross-sections of a bundle, Proc. Amer. Math. Soc. 103 (1988), 956-960. MR 947690 (90e:57036)
  • [Sa$ _{2}$] -, The space of Lipschitz maps from a compactum to an absolute neighborhood LIP extensor, preprint.
  • [SW$ _{1}$] K. Sakai and R. Y. Wong, The space of Lipschitz maps from a compactum to a locally convex set, Topology Appl., 32 (1989), 223-235. MR 1007102 (90j:57013)
  • [SW$ _{2}$] -, On infinite-dimensional manifold triples, Trans. Amer. Math. Soc., (to appear). MR 994171 (90g:57017)
  • [To] H. Toruńczyk, Concerning locally homotopy negligible sets and characterization of $ {l_2}$-manifolds, Fund. Math. 101 (1978), 93-110. MR 518344 (80g:57019)
  • [Wo] R. Y. Wong, Lipschitz conjugation and extension of homeomorphisms in $ {l_p}$-spaces, J. Math. Analysis Appl. 32 (1970), 573-583. MR 0273391 (42:8270)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57N20, 54C35, 58B05, 58D15

Retrieve articles in all journals with MSC: 57N20, 54C35, 58B05, 58D15


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

American Mathematical Society