In this paper we develop the semifilter approach to the classical Menger and Hurewicz properties and show that the small cardinal $𝔤$ is a lower bound of the additivity number of the $\sigma$-ideal generated by Menger subspaces of the Baire space, and under $𝔲<𝔤$ every subset $X$ of the real line with the property $Split(\Lambda ,\Lambda )$ is Hurewicz, and thus it is consistent with ZFC that the property $Split(\Lambda ,\Lambda )$ is preserved by unions of less than $𝔟$ subsets of the real line.

Developing the idea of assigning to a large cover of a topological space a corresponding semifilter, we show that every Menger topological space has the property ${\bigcup }_{fin}(𝒪,T^{*})$ provided $(𝔲<𝔤)$, and every space with the property ${\bigcup }_{fin}(𝒪,T^{*})$ is Hurewicz provided $({Depth}^{+}\left({\left[\omega \right]}^{{\aleph }_0}\right)\le 𝔟)$. Combining this with the results proven in cited literature, we settle all questions whether (it is consistent that) the properties $\text{P}$ and $\text{Q}$ [do not] coincide, where $\text{P}$ and $\text{Q}$ run over ${\bigcup }_{fin}(𝒪,\Gamma )$, ${\bigcup }_{fin}(𝒪,T)$, ${\bigcup }_{fin}(𝒪,T^{*})$, ${\bigcup }_{fin}(𝒪,\Omega )$, and ${\bigcup }_{fin}(𝒪,𝒪)$.

This short note shows that the scheme of disjunctive reasoning, a or b, not b : a, does not hold neither in proper ortholattices nor in proper de Morgan algebras. In both cases the scheme, once translated into the inequality b' · (a+b) ≤ a, forces the structure to be a boolean algebra.

In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $\alpha \le 2^{}$, a topological group G such that $G^{\gamma }$ is countably compact for all cardinals γ < α, but $G^{\alpha }$ is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $M{A}_{countable}$. Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M{A}_{countable}$. However, the question has remained...

Using the method of forcing we prove that consistently there is a Banach space (of continuous functions on a totally disconnected compact Hausdorff space) of density κ bigger than the continuum where all operators are multiplications by a continuous function plus a weakly compact operator and which has no infinite-dimensional complemented subspaces of density continuum or smaller. In particular no separable infinite-dimensional subspace has a complemented superspace of density continuum or smaller,...

MV-algebras were introduced by Chang, 1958 as algebraic bases for multi-valued logic. MV stands for “multi-valued" and MV algebras have already occupied an important place in the realm of nonstandard (mathematical) logic applied in several fields including cybernetics. In the present paper, using the Loomis–Sikorski theorem for $\sigma$-MV-algebras, we prove that, with every element $a$ in a $\sigma$-MV algebra $M$, a spectral measure (i. e. an observable) ${\Lambda }_a:ℬ\left([0,1]\right)\to ℬ\left(M\right)$ can be associated, where $ℬ\left(M\right)$ denotes the Boolean $\sigma$-algebra...