Locallyn-Connected Compacta and UV n -Maps

V. Valov

Analysis and Geometry in Metric Spaces (2015)

  • Volume: 3, Issue: 1, page 93-101, electronic only
  • ISSN: 2299-3274

Abstract

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We provide a machinery for transferring some properties of metrizable ANR-spaces to metrizable LCn-spaces. As a result, we show that for completely metrizable spaces the properties ALCn, LCn and WLCn coincide to each other. We also provide the following spectral characterizations of ALCn and celllike compacta: A compactum X is ALCn if and only if X is the limit space of a σ-complete inverse system S = {Xα , pβ α , α < β < τ} consisting of compact metrizable LCn-spaces Xα such that all bonding projections pβα, as a well all limit projections pα, are UVn-maps. A compactum X is a cell-like (resp., UVn) space if and only if X is the limit space of a σ-complete inverse system consisting of cell-like (resp., UVn) metrizable compacta.

How to cite

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V. Valov. "Locallyn-Connected Compacta and UV n -Maps." Analysis and Geometry in Metric Spaces 3.1 (2015): 93-101, electronic only. <http://eudml.org/doc/271082>.

@article{V2015,
abstract = {We provide a machinery for transferring some properties of metrizable ANR-spaces to metrizable LCn-spaces. As a result, we show that for completely metrizable spaces the properties ALCn, LCn and WLCn coincide to each other. We also provide the following spectral characterizations of ALCn and celllike compacta: A compactum X is ALCn if and only if X is the limit space of a σ-complete inverse system S = \{Xα , pβ α , α < β < τ\} consisting of compact metrizable LCn-spaces Xα such that all bonding projections pβα, as a well all limit projections pα, are UVn-maps. A compactum X is a cell-like (resp., UVn) space if and only if X is the limit space of a σ-complete inverse system consisting of cell-like (resp., UVn) metrizable compacta.},
author = {V. Valov},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {absolute neighborhood retracts; ALCn-spaces; cell-like maps and spaces; WLCn-spaces; UVn-maps and space; ALC-spaces; WLC-spaces; UV-maps and spaces},
language = {eng},
number = {1},
pages = {93-101, electronic only},
title = {Locallyn-Connected Compacta and UV n -Maps},
url = {http://eudml.org/doc/271082},
volume = {3},
year = {2015},
}

TY - JOUR
AU - V. Valov
TI - Locallyn-Connected Compacta and UV n -Maps
JO - Analysis and Geometry in Metric Spaces
PY - 2015
VL - 3
IS - 1
SP - 93
EP - 101, electronic only
AB - We provide a machinery for transferring some properties of metrizable ANR-spaces to metrizable LCn-spaces. As a result, we show that for completely metrizable spaces the properties ALCn, LCn and WLCn coincide to each other. We also provide the following spectral characterizations of ALCn and celllike compacta: A compactum X is ALCn if and only if X is the limit space of a σ-complete inverse system S = {Xα , pβ α , α < β < τ} consisting of compact metrizable LCn-spaces Xα such that all bonding projections pβα, as a well all limit projections pα, are UVn-maps. A compactum X is a cell-like (resp., UVn) space if and only if X is the limit space of a σ-complete inverse system consisting of cell-like (resp., UVn) metrizable compacta.
LA - eng
KW - absolute neighborhood retracts; ALCn-spaces; cell-like maps and spaces; WLCn-spaces; UVn-maps and space; ALC-spaces; WLC-spaces; UV-maps and spaces
UR - http://eudml.org/doc/271082
ER -

References

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  1. [1] P. Bacon, Extending a complete metric, Amer. Math. Monthly 75 (1968) 642-643.  Zbl0159.52603
  2. [2] S. Bogatyi and Ju. Smirnov, Approximation by polyhedra and factorization theorems for ANR-bicompacta. Fund. Math. 87 (1975), no. 3, 195–205 (in Russian).  Zbl0305.54012
  3. [3] A. Chigogidze, Inverse spectra, North-Holland Math. Library 53 (Elsevier Sci. B.V., Amsterdam, 1996)  
  4. [4] A. Chigogidze, The theory of n-shapes, Russian Math. Surveys 44 (1989), 145–174.  Zbl0696.54013
  5. [5] A. Dranishnikov, A private communication 1990.  
  6. [6] A. Dranishnikov, Universal Menger compacta and universal mappings, Math. USSR Sb. 57 (1987), no. 1, 131–149.  Zbl0622.54026
  7. [7] A. Dranishnikov, Absolute ekstensors in dimension n and n-soft maps increasing dimension, UspekhiMat. Nauk 39 (1984), no. 5(239), 55–95 (in Russian).  
  8. [8] J. Dugundji, Modified Vietoris theorem for homotopy, Fund. Math. 66 (1970), 223–235.  Zbl0196.26801
  9. [9] J. Dugundji and E. Michael, On local and uniformly local topological properties, Proc. Amer. Math. Soc. 7 (1956), 304–307.  Zbl0071.38203
  10. [10] V. Gutev, Selections for quasi-l.sc. mappings with uniformly equi-LCn range, Set-Valued Anal. 1 (1993), no. 4, 319–328.  Zbl0809.54016
  11. [11] S. Mardešic, On covering dimension and inverse limits of compact spaces, Illinois Math. Joourn. 4 (1960), no. 2, 278–291.  Zbl0094.16902
  12. [12] N. To Nhu, Investigating the ANR-property of metric spaces, Fund. Math. 124 (1984), 243–254; Corrections: Fund. Math. 141 (1992), 297.  Zbl0573.54009
  13. [13] S. T. Hu, Theory of retracts, Wayne State Univ. Press, Detroit, 1965.  Zbl0145.43003
  14. [14] A. Karassev and V. Valov, Extension dimension and quasi-finite CW-complexes, Topology Appl. 153 (2006), 3241–3254. [WoS] Zbl1177.54016
  15. [15] B. Pasynkov, Monotonicity of dimension and open mappings that raise dimension, Trudy Mat. Inst. Steklova 247 (2004), 202–213 (in Russian).  
  16. [16] B. Pasynkov, On universal bicompacta of given weight and dimension, Dokl. Akad. Nauk SSSR 154 (1964), no. 5, 1042–1043 (in Russian).  Zbl0197.48601
  17. [17] K. Sakai, Geometric aspects of general topology, Springer Monographs in Mathematics. Springer, Tokyo, 2013.  Zbl1280.54001
  18. [18] E. Shchepin, Topology of limit spaces with uncountable inverse spectra, Uspekhi Mat. Nauk 31 (1976), no. 5(191), 191–226 (in Russian).  

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