×

\({\mathcal R}\)-compact spaces with \(\text{weight }X < \text{Exp}_{\mathcal R}X\). (English) Zbl 0977.54022

It is a classical result of Tikhonov that a completely regular space of weight \(\kappa\) can be embedded into \(\mathbb{R}^\kappa\). As witnessed by the space \(\mathbb{Q}\) of rational numbers it need not be the case that a realcompact space of weight \(\kappa\) may be embedded into \(\mathbb{R}^\kappa\) as a closed subspace. For a realcompact \(X\) the author uses \(\text{Exp}_\mathcal{R}X\) to denote the smallest \(\kappa\) such that \(X\) embeds as a closed subspace of \(\mathbb{R}^\kappa\). So, in this notation, \(\text{weight }\mathbb{Q}<\text{Exp}_\mathcal{R}\mathbb{Q}\). The author uses an analogue of \(\mathbb{Q}\) to show that this phenomenon persists in higher powers. For a cardinal \(\kappa\) let \(\mathbb{Q}(\kappa)\) be the subset of \(2^\kappa\) consisting of the points whose supports have cardinality less than \(\kappa\) and with the topology generated by the open boxes whose supports have cardinality less than \(\kappa\). If \(\kappa=\lambda^+=2^\lambda\) then \(\mathbb{Q}(\kappa)\) is an example of a hereditarily \(\mathbb{N}\)-compact space with \(\text{weight }\mathbb{Q}(\kappa)=\kappa\) and \(\text{Exp}_\mathcal{R}\mathbb{Q}>\kappa\).
The paper closes with discussions and questions on the possibility of obtaining examples as above in ZFC.
Reviewer: K.P.Hart (Delft)

MSC:

54D60 Realcompactness and realcompactification
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54G20 Counterexamples in general topology
54B10 Product spaces in general topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111 – 167. · Zbl 0561.54004
[2] S.H. Heckler, Exponents of some \(\mathscr{N}\)-compact spaces, Israel J. Math. 15 (1973), 384 - 395.
[3] John Kulesza, Metrizable spaces where the inductive dimensions disagree, Trans. Amer. Math. Soc. 318 (1990), no. 2, 763 – 781. · Zbl 0724.54034
[4] S. Mrówka, On \?-compact spaces. II, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14 (1966), 597 – 605 (English, with Russian summary). · Zbl 0161.19603
[5] S. Mrówka, Further results on \?-compact spaces. I, Acta Math. 120 (1968), 161 – 185. · Zbl 0179.51202 · doi:10.1007/BF02394609
[6] S. Mrówka, Extending of continuous real functions, Compositio Math. 21 (1969), 319 – 327. · Zbl 0187.44503
[7] Stanislaw Mrowka, \?-compactness, metrizability, and covering dimension, Rings of continuous functions (Cincinnati, Ohio, 1982) Lecture Notes in Pure and Appl. Math., vol. 95, Dekker, New York, 1985, pp. 247 – 275. · Zbl 0564.54015
[8] -, Preservation of \(E\)-compactness, in preparation. · Zbl 0332.54021
[9] S. Mrówka, Small inductive dimension of completions of metric spaces, Proc. Amer. Math. Soc. 125 (1997), no. 5, 1545 – 1554. · Zbl 0870.54033
[10] R. Sikorski, Remarks on some topological spaces of high power, Fund. Math. 37 (1950), 125 - 136. · Zbl 0041.09705
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.