# Article

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Keywords:
alternative set theory; biequivalence vector space; $\pi$-equivalence; continuous function; set uniform equivalence; compact; dimensionally compact
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.
References:
[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
[K-Z 1988] Kalina M., Zlatoš P.: Arithmetic of cuts and cuts of classes. Comment. Math. Univ. Carolinae 29 (1988), 435-456. MR 0972828
[M 1979] Mlček J.: Valuations of structures. Comment. Math. Univ. Carolinae 20 (1979), 681-696. MR 0555183
[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.
[Sm 1987] Šmíd M.: personal communication.
[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
[V 1979] Vopěnka P.: Mathematics in the Alternative Set Theory. Teubner-Verlag Leipzig. MR 0581368
[V 1979a] Vopěnka P.: The lattice of indiscernibility equivalences. Comment. Math. Univ. Carolinae 20 (1979), 631-638. MR 0555179
[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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