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Keywords:
alternative set theory; nonstandard analysis; biequivalence vector space; compact; dimensionally compact; indiscernibles; Ramsey theorem
alternative set theory; nonstandard analysis; biequivalence vector space; compact; dimensionally compact; indiscernibles; Ramsey theorem
Summary:
This is a contribution to the theory of topological vector spaces within the framework of the alternative set theory. Using indiscernibles we will show that every infinite set $u S\subseteq G$ in a biequivalence vector space $\langle W,M,G\rangle$, such that $x - y \notin M$ for distinct $x,y \in u$, contains an infinite independent subset. Consequently, a class $X \subseteq G$ is dimensionally compact iff the $\pi$-equivalence $\doteq_M$ is compact on $X$. This solves a problem from the paper [NPZ 1992] by J. Náter, P. Pulmann and the second author.
References:
[{GZ 1985}] Guričan J., Zlatoš P.: Biequivalences and topology in the alternative set theory. Comment. Math. Univ. Carolinae 26.3 525-552. MR 0817825
[{NPZ 1992}] Náter J., Pulmann P., Zlatoš P.: Dimensional compactness in biequivalence vector spaces. Comment. Math. Univ. Carolinae 33.4 681-688. MR 1240189
[{ŠZ 1991}] Šmíd M., Zlatoš P.: Biequivalence vector spaces in the alternative set theory. Comment. Math. Univ. Carolinae 32.3 517-544. MR 1159799
[{SVe 1981}] Sochor A., Vencovská A.: Indiscernibles in the alternative set theory. Comment. Math. Univ. Carolinae 22.4 785-798. MR 0647026
[{V 1979}] Vopěnka P.: Mathematics in the Alternative Set Theory. Teubner-Verlag Leipzig. MR 0581368
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