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Notions of flatness relative to a Grothendieck topology.

Karazeris, Panagis (2004)

Theory and Applications of Categories [electronic only]

Novak's result by Henkin's method

H. Doest (1969)

Fundamenta Mathematicae

Nowhere dense subsets and Booth's Lemma

Viacheslav I. Malykhin (1996)

Commentationes Mathematicae Universitatis Carolinae

The following statement is proved to be independent from $[LB + \neg CH]$ : $(*)$ Let $X$ be a Tychonoff space with $c (X) \leq ℵ_{0}$ and $π w (X) < ℭ$ . Then a union of less than $ℭ$ of nowhere dense subsets of $X$ is a union of not greater than $π w (X)$ of nowhere dense subsets.

Null events and stochastical independence

Giulianella Colleti, Romano Scozzafava (1998)

Kybernetika

In this paper we point out the lack of the classical definitions of stochastical independence (particularly with respect to events of 0 and 1 probability) and then we propose a definition that agrees with all the classical ones when the probabilities of the relevant events are both different from 0 and 1, but that is able to focus the actual stochastical independence also in these extreme cases. Therefore this definition avoids inconsistencies such as the possibility that an event $A$ can be at the same time stochastically independent and logically dependent on an event $B$ . In a forthcoming paper we will deepen (in this context) the concept of conditional independence (which is just sketched in the last section of the present paper) and we will deal also with the extension of these results to the general case of any (finite) number of events.