# Article

 Title: Dimensional compactness in biequivalence vector spaces (English) Author: Náter, J. Author: Pulmann, P. Author: Zlatoš, P. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 33 Issue: 4 Year: 1992 Pages: 681-688 . Category: math . Summary: The notion of dimensionally compact class in a biequivalence vector space is introduced. Similarly as the notion of compactness with respect to a $\pi$-equivalence reflects our nonability to grasp any infinite set under sharp distinction of its elements, the notion of dimensional compactness is related to the fact that we are not able to measure out any infinite set of independent parameters. A fairly natural Galois connection between equivalences on an infinite set $s$ and classes of set functions $s \rightarrow Q$ is investigated. Finally, a direct connection between compactness of a $\pi$-equivalence $R \subseteq s^2$ and dimensional compactness of the class $\bold C[R]$ of all continuous set functions from $\langle s,R \rangle$ to $\langle Q,\doteq \rangle$ is established. (English) Keyword: alternative set theory Keyword: biequivalence vector space Keyword: $\pi$-equivalence Keyword: continuous function Keyword: set uniform equivalence Keyword: compact Keyword: dimensionally compact MSC: 03E70 MSC: 03H05 MSC: 46E25 MSC: 46S10 MSC: 46S20 MSC: 46S99 idZBL: Zbl 0784.46064 idMR: MR1240189 . Date available: 2009-01-08T17:59:40Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/118539 . Reference: [G-Z 1985] Guričan J., Zlatoš P.: Biequivalences and topology in the alternative set theory.Comment. Math. Univ. Carolinae 26 (1985), 525-552. MR 0817825 Reference: [K-Z 1988] Kalina M., Zlatoš P.: Arithmetic of cuts and cuts of classes.Comment. Math. Univ. Carolinae 29 (1988), 435-456. MR 0972828 Reference: [M 1979] Mlček J.: Valuations of structures.Comment. Math. Univ. Carolinae 20 (1979), 681-696. MR 0555183 Reference: [M 1990] Mlček J.: Some structural and combinatorial properties of classes in the alternative set theory (in Czech).habilitation Faculty of Mathematics and Physics, Charles University Prague. Reference: [Sm 1987] Šmíd M.: personal communication.. Reference: [Sm-Z 1991] Šmíd M., Zlatoš P.: Biequivalence vector spaces in the alternative set theory.Comment. Math. Univ. Carolinae 32 (1991), 517-544. MR 1159799 Reference: [V 1979] Vopěnka P.: Mathematics in the Alternative Set Theory.Teubner-Verlag Leipzig. MR 0581368 Reference: [V 1979a] Vopěnka P.: The lattice of indiscernibility equivalences.Comment. Math. Univ. Carolinae 20 (1979), 631-638. MR 0555179 Reference: [Z 1989] P. Zlatoš: Topological shapes.Proc. of the 1st Symposium on Mathematics in the Alternative Set Theory J. Mlček et al. Association of Slovak Mathematicians and Physicists Bratislava 95-120. .

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