Geodesic
Dimensional compactness in biequivalence vector spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) no. 4, pp. 681-688 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 03E70, 03H05, 46E25, 46S10, 46S20, 46S99
Keywords: alternative set theory; biequivalence vector space; $\pi$-equivalence; continuous function; set uniform equivalence; compact; dimensionally compact 
@article{CMUC_1992_33_4_a12,
     author = {N\'ater, J. and Pulmann, P. and Zlato\v{s}, P.},
     title = {Dimensional compactness in biequivalence vector spaces},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {681--688},
     year = {1992},
     volume = {33},
     number = {4},
     mrnumber = {1240189},
     zbl = {0784.46064},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1992_33_4_a12/}
}
TY  - JOUR
AU  - Náter, J.
AU  - Pulmann, P.
AU  - Zlatoš, P.
TI  - Dimensional compactness in biequivalence vector spaces
JO  - Commentationes Mathematicae Universitatis Carolinae
PY  - 1992
SP  - 681
EP  - 688
VL  - 33
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CMUC_1992_33_4_a12/
LA  - en
ID  - CMUC_1992_33_4_a12
ER  - 
%0 Journal Article
%A Náter, J.
%A Pulmann, P.
%A Zlatoš, P.
%T Dimensional compactness in biequivalence vector spaces
%J Commentationes Mathematicae Universitatis Carolinae
%D 1992
%P 681-688
%V 33
%N 4
%U http://geodesic.mathdoc.fr/item/CMUC_1992_33_4_a12/
%G en
%F CMUC_1992_33_4_a12
Náter, J.; Pulmann, P.; Zlatoš, P. Dimensional compactness in biequivalence vector spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 33 (1992) no. 4, pp. 681-688. http://geodesic.mathdoc.fr/item/CMUC_1992_33_4_a12/

[G-Z 1985] Guričan J., Zlatoš P.: Biequivalences and topology in the alternative set theory. Comment. Math. Univ. Carolinae 26 (1985), 525-552. | MR

[K-Z 1988] Kalina M., Zlatoš P.: Arithmetic of cuts and cuts of classes. Comment. Math. Univ. Carolinae 29 (1988), 435-456. | MR

[M 1979] Mlček J.: Valuations of structures. Comment. Math. Univ. Carolinae 20 (1979), 681-696. | MR

[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

[V 1979] Vopěnka P.: Mathematics in the Alternative Set Theory. Teubner-Verlag Leipzig. | MR

[V 1979a] Vopěnka P.: The lattice of indiscernibility equivalences. Comment. Math. Univ. Carolinae 20 (1979), 631-638. | MR

[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.