There exists a natural extension of the notion of preorder from binary relations onto relations whose arities are arbitrary ordinals. In the article we find a condition under which extended preorders coincide with preorders if viewed categorically.
We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space is countably compact if and only if it is countably subcompact relative to . (iii) For every metrizable space , the following are equivalent: (a) is compact; (b) for every open filter of , ; (c) is subcompact relative to . We also show: (iv) The negation of each of the statements, (a) every countably subcompact metrizable...
In this short note we show that for fuzzy sets of type II the additive rule for cardinalities holds true. The proof of this result requires an application of approximate reasoning as means of inference by use of the entailment principle.
The cardinal invariants of are known to satisfy that . We prove that all inequalities can be strict. We also introduce a new upper bound for and show that it can be less than . The key method is to utilize finite support matrix iterations of ccc posets following paper Ultrafilters with small generating sets by A. Blass and S. Shelah (1989).
A probability theory on IFS-events has been constructed in [3], and axiomatically characterized in [4]. Here using a general system of axioms it is shown that any probability on IFS-events can be decomposed onto two probabilities on a Lukasiewicz tribe, hence some known results from [5], [6] can be used also for IFS-sets. As an application of the approach a variant of Central limit theorem is presented.
In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements " is countably compact" and " is compact"