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Dimensional compactness in biequivalence vector spaces. (English) Zbl 0784.46064

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 \(s\to Q\) is investigated. Finally, a direct connection between compactness of a \(\pi\)-equivalence \(R\subseteq s^ 2\) and dimensional compactness of the class \(C[R]\) of all continuous set functions from \(\langle s,R\rangle\) to \(\langle Q,\doteq\rangle\) is established.

MSC:

46S20 Nonstandard functional analysis
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46E25 Rings and algebras of continuous, differentiable or analytic functions
03E70 Nonclassical and second-order set theories
03H05 Nonstandard models in mathematics
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