On the disjoint (0,N)-cells property for homogeneous ANR's

Paweł Krupski

Colloquium Mathematicae (1993)

  • Volume: 66, Issue: 1, page 77-84
  • ISSN: 0010-1354

Abstract

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A metric space (X,ϱ) satisfies the disjoint (0,n)-cells property provided for each point x ∈ X, any map f of the n-cell B n into X and for each ε > 0 there exist a point y ∈ X and a map g : B n → X such that ϱ(x,y) < ε, ϱ ^ ( f , g ) < ε and y ∉ g ( B n ) . It is proved that each homogeneous locally compact ANR of dimension >2 has the disjoint (0,2)-cells property. If dimX = n > 0, X has the disjoint (0,n-1)-cells property and X is a locally compact L C n - 1 -space then local homologies satisfy H k ( X , X - x ) = 0 for k < n and Hn(X,X-x) ≠ 0.

How to cite

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Krupski, Paweł. "On the disjoint (0,N)-cells property for homogeneous ANR's." Colloquium Mathematicae 66.1 (1993): 77-84. <http://eudml.org/doc/210236>.

@article{Krupski1993,
abstract = {A metric space (X,ϱ) satisfies the disjoint (0,n)-cells property provided for each point x ∈ X, any map f of the n-cell $B^\{n\}$ into X and for each ε > 0 there exist a point y ∈ X and a map $g:B^\{n\} → X$ such that ϱ(x,y) < ε, $\widehat\{ϱ\}(f,g) < ε$ and $y ∉ g(B^\{n\})$. It is proved that each homogeneous locally compact ANR of dimension >2 has the disjoint (0,2)-cells property. If dimX = n > 0, X has the disjoint (0,n-1)-cells property and X is a locally compact $LC^\{n-1\}$-space then local homologies satisfy $H_\{k\}(X,X-x) = 0$ for k < n and Hn(X,X-x) ≠ 0.},
author = {Krupski, Paweł},
journal = {Colloquium Mathematicae},
keywords = {generalized manifold; homogeneous space; disjoint cells property; absolute neighborhood retract; $LC^n$-space; -space},
language = {eng},
number = {1},
pages = {77-84},
title = {On the disjoint (0,N)-cells property for homogeneous ANR's},
url = {http://eudml.org/doc/210236},
volume = {66},
year = {1993},
}

TY - JOUR
AU - Krupski, Paweł
TI - On the disjoint (0,N)-cells property for homogeneous ANR's
JO - Colloquium Mathematicae
PY - 1993
VL - 66
IS - 1
SP - 77
EP - 84
AB - A metric space (X,ϱ) satisfies the disjoint (0,n)-cells property provided for each point x ∈ X, any map f of the n-cell $B^{n}$ into X and for each ε > 0 there exist a point y ∈ X and a map $g:B^{n} → X$ such that ϱ(x,y) < ε, $\widehat{ϱ}(f,g) < ε$ and $y ∉ g(B^{n})$. It is proved that each homogeneous locally compact ANR of dimension >2 has the disjoint (0,2)-cells property. If dimX = n > 0, X has the disjoint (0,n-1)-cells property and X is a locally compact $LC^{n-1}$-space then local homologies satisfy $H_{k}(X,X-x) = 0$ for k < n and Hn(X,X-x) ≠ 0.
LA - eng
KW - generalized manifold; homogeneous space; disjoint cells property; absolute neighborhood retract; $LC^n$-space; -space
UR - http://eudml.org/doc/210236
ER -

References

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  1. [1] R. H. Bing and K. Borsuk, Some remarks concerning topologically homogeneous spaces, Ann. of Math. 81 (1965), 100-111. Zbl0127.13302
  2. [2] K. Borsuk, Theory of Retracts, PWN-Polish Sci. Publ., Warszawa, 1967. Zbl0153.52905
  3. [3] J. J. Charatonik and T. Maćkowiak, Around Effros' theorem, Trans. Amer. Math. Soc. 298 (1986), 579-602. Zbl0608.54012
  4. [4] R. J. Daverman, Detecting the disjoint disks property, Pacific J. Math. 93 (1981), 277-298. Zbl0415.57007
  5. [5] R. J. Daverman, Decompositions of Manifolds, Academic Press, Orlando, 1986. Zbl0608.57002
  6. [6] P. Krupski, Homogeneity and Cantor manifolds, Proc. Amer. Math. Soc. 109 (1990), 1135-1142. Zbl0714.54035
  7. [7] P. Krupski, Recent results on homogeneous curves and ANR's, Topology Proc. 16 (1991), 109-118. Zbl0801.54015
  8. [8] J. M. Łysko, On homogeneous ANR-spaces, in: Proc. Internat. Conf. on Geometric Topology, PWN-Polish Sci. Publ., Warszawa, 1980, 305-306. 
  9. [9] J. van Mill, Infinite-Dimensional Topology, North-Holland, Amsterdam, 1989. 
  10. [10] W. J. R. Mitchell, General position properties of ANR's, Math. Proc. Cambridge Philos. Soc. 92 (1982), 451-466. Zbl0529.57008
  11. [11] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. 

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