Approximate inverse systems of uniform spaces and an application of inverse systems

Michael G. Charalambous

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 3, page 551-565
  • ISSN: 0010-2628

Abstract

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The fundamental properties of approximate inverse systems of uniform spaces are established. The limit space of an approximate inverse sequence of complete metric spaces is the limit of an inverse sequence of some of these spaces. This has an application to the dimension of the limit space of an approximate inverse system. A topologically complete space with dim ≤ n is the limit of an approximate inverse system of metric polyhedra of dim ≤ n . A completely metrizable separable space with dim ≤ n is the limit of an inverse sequence of locally finite polyhedra of dim ≤ n . Finally, a new proof is derived of the important equality dim = Ind for metric spaces.

How to cite

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Charalambous, Michael G.. "Approximate inverse systems of uniform spaces and an application of inverse systems." Commentationes Mathematicae Universitatis Carolinae 32.3 (1991): 551-565. <http://eudml.org/doc/247292>.

@article{Charalambous1991,
abstract = {The fundamental properties of approximate inverse systems of uniform spaces are established. The limit space of an approximate inverse sequence of complete metric spaces is the limit of an inverse sequence of some of these spaces. This has an application to the dimension of the limit space of an approximate inverse system. A topologically complete space with $\operatorname\{dim\} \le n$ is the limit of an approximate inverse system of metric polyhedra of $\operatorname\{dim\} \le n$. A completely metrizable separable space with $\operatorname\{dim\} \le n$ is the limit of an inverse sequence of locally finite polyhedra of $\operatorname\{dim\} \le n$. Finally, a new proof is derived of the important equality $\operatorname\{dim\} = \operatorname\{Ind\}$ for metric spaces.},
author = {Charalambous, Michael G.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {inverse systems; approximate inverse systems; uniform; metric and complete spaces; covering and inductive dimension; covering dimension; inductive dimension; approximate inverse systems; uniform spaces; complete metric spaces; inverse sequence; complete space},
language = {eng},
number = {3},
pages = {551-565},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Approximate inverse systems of uniform spaces and an application of inverse systems},
url = {http://eudml.org/doc/247292},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Charalambous, Michael G.
TI - Approximate inverse systems of uniform spaces and an application of inverse systems
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 3
SP - 551
EP - 565
AB - The fundamental properties of approximate inverse systems of uniform spaces are established. The limit space of an approximate inverse sequence of complete metric spaces is the limit of an inverse sequence of some of these spaces. This has an application to the dimension of the limit space of an approximate inverse system. A topologically complete space with $\operatorname{dim} \le n$ is the limit of an approximate inverse system of metric polyhedra of $\operatorname{dim} \le n$. A completely metrizable separable space with $\operatorname{dim} \le n$ is the limit of an inverse sequence of locally finite polyhedra of $\operatorname{dim} \le n$. Finally, a new proof is derived of the important equality $\operatorname{dim} = \operatorname{Ind}$ for metric spaces.
LA - eng
KW - inverse systems; approximate inverse systems; uniform; metric and complete spaces; covering and inductive dimension; covering dimension; inductive dimension; approximate inverse systems; uniform spaces; complete metric spaces; inverse sequence; complete space
UR - http://eudml.org/doc/247292
ER -

References

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  1. Charalambous M.G., A new covering dimension function for uniform spaces, J. London Math. Soc. (2) 11 (1975), 137-143. (1975) Zbl0306.54048MR0375258
  2. Charalambous M.G., The dimension of inverse limits, Proc. Amer. 58 (1976), 289-295. (1976) Zbl0348.54029MR0410696
  3. Engelking R., General Topology, Polish Scientific Publishers, Warsaw, 1977. Zbl0684.54001MR0500780
  4. Engelking R., Dimension Theory, Polish Scientific Publishers, Warsaw, 1978. Zbl0401.54029MR0482697
  5. Freudenthal H., Entwicklungen von Räumen und ihren Gruppen, Compositio Math. 4 (1937), 145-234. (1937) Zbl0016.28001MR1556968
  6. Hurewicz W., Wallman H., Dimension Theory, Princeton University Press, Princeton, 1941. Zbl0036.12501MR0006493
  7. Isbell J.R., Uniform spaces, Amer. Math. Soc. Surveys 12, 1964. Zbl0124.15601MR0170323
  8. Mardešić S., On covering dimension and inverse limits of compact spaces, Illinois J. Math. 4 (1960), 278-291. (1960) MR0116306
  9. Mardešić S., Rubin L.R., Approximate uniform spaces of compacta and covering dimension, Pacific J. Math. 138 (1989), 129-144. (1989) MR0992178
  10. Nagami K., Dimension Theory, Academic Press, New York and London, 1970. Zbl0224.54060MR0271918
  11. Pasynkov B.A., On polyhedra spectra and dimension of bicompacta and of bicompact groups (in Russian), Dokl. Akad. Nauk SSSR 121 (1958), 45-48. (1958) MR0102058
  12. Pasynkov B.A., Factorization theorems in dimension theory, Russian Math. Surveys 36 (1981), 175-209. (1981) Zbl0487.54034MR0622723
  13. Pears A.R., Dimension Theory of General Spaces, Cambridge Univ. Press, Cambridge, 1976. Zbl0312.54001MR0394604
  14. Spanier E.H., Algebraic Topology, McGraw-Hill Inc., New York, 1966. Zbl0810.55001MR0210112

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