Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
inverse systems; approximate inverse systems; uniform; metric and complete spaces; covering and inductive dimension
inverse systems; approximate inverse systems; uniform; metric and complete spaces; covering and inductive dimension
Summary:
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 $\operatorname{dim} \leq n$ is the limit of an approximate inverse system of metric polyhedra of $\operatorname{dim} \leq n$. A completely metrizable separable space with $\operatorname{dim} \leq n$ is the limit of an inverse sequence of locally finite polyhedra of $\operatorname{dim} \leq n$. Finally, a new proof is derived of the important equality $\operatorname{dim} = \operatorname{Ind}$ for metric spaces.
References:
[1] Charalambous M.G.: A new covering dimension function for uniform spaces. J. London Math. Soc. (2) 11 (1975), 137-143. MR 0375258 | Zbl 0306.54048
[2] Charalambous M.G.: The dimension of inverse limits. Proc. Amer. 58 (1976), 289-295. MR 0410696 | Zbl 0348.54029
[3] Engelking R.: General Topology. Polish Scientific Publishers, Warsaw, 1977. MR 0500780 | Zbl 0684.54001
[4] Engelking R.: Dimension Theory. Polish Scientific Publishers, Warsaw, 1978. MR 0482697 | Zbl 0401.54029
[5] Freudenthal H.: Entwicklungen von Räumen und ihren Gruppen. Compositio Math. 4 (1937), 145-234. MR 1556968 | Zbl 0016.28001
[6] Hurewicz W., Wallman H.: Dimension Theory. Princeton University Press, Princeton, 1941. MR 0006493 | Zbl 0036.12501
[7] Isbell J.R.: Uniform spaces. Amer. Math. Soc. Surveys 12, 1964. MR 0170323 | Zbl 0124.15601
[8] Mardešić S.: On covering dimension and inverse limits of compact spaces. Illinois J. Math. 4 (1960), 278-291. MR 0116306
[9] Mardešić S., Rubin L.R.: Approximate uniform spaces of compacta and covering dimension. Pacific J. Math. 138 (1989), 129-144. MR 0992178
[10] Nagami K.: Dimension Theory. Academic Press, New York and London, 1970. MR 0271918 | Zbl 0224.54060
[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. MR 0102058
[12] Pasynkov B.A.: Factorization theorems in dimension theory. Russian Math. Surveys 36 (1981), 175-209. MR 0622723 | Zbl 0487.54034
[13] Pears A.R.: Dimension Theory of General Spaces. Cambridge Univ. Press, Cambridge, 1976. MR 0394604 | Zbl 0312.54001
[14] Spanier E.H.: Algebraic Topology. McGraw-Hill Inc., New York, 1966. MR 0210112 | Zbl 0810.55001
Search
Partner of
