We say that a real normed lattice is quasi-Baire if the intersection of each sequence of monotonic open dense sets is dense. An example of a Baire-convex space, due to M. Valdivia, which is not quasi-Baire is given. We obtain that $E$ is a quasi-Baire space iff $(E,T\left(𝒰\right),T\left({𝒰}^{-1}\right))$, is a pairwise Baire bitopological space, where $𝒰$, is a quasi-uniformity that determines, in $L$. Nachbin’s sense, the topological ordered space $E$.

We give a proof of a theorem of Maćkowiak on the existence of universal n-dimensional hereditarily indecomposable continua, based on the Baire-category method.

Let G be a locally compact group with a fixed left Haar measure. Given Young functions φ and ψ, we consider the Orlicz spaces $L^{\phi }\left(G\right)$ and $L^{\psi }\left(G\right)$ on a non-unimodular group G, and, among other things, we prove that under mild conditions on φ and ψ, the set $(f,g)\in L^{\phi }\left(G\right)\times L^{\psi }\left(G\right):f*g$ is well defined on G is σ-c-lower porous in $L^{\phi }\left(G\right)\times L^{\psi }\left(G\right)$. This answers a question raised by Głąb and Strobin in 2010 in a more general setting of Orlicz spaces. We also prove a similar result for non-compact locally compact groups.

We investigate an algebraic notion of decidability which allows a uniform investigation of a large class of notions of forcing. Among other things, we show how to build σ-fields of sets connected with Laver and Miller notions of forcing and we show that these σ-fields are closed under the Suslin operation.

For a non-isolated point $x$ of a topological space $X$ let ${\mathrm{nw}}_{\chi }\left(x\right)$ be the smallest cardinality of a family $𝒩$ of infinite subsets of $X$ such that each neighborhood $O\left(x\right)\subset X$ of $x$ contains a set $N\in 𝒩$. We prove that (a) each infinite compact Hausdorff space $X$ contains a non-isolated point $x$ with ${\mathrm{nw}}_{\chi }\left(x\right)={\aleph }_0$; (b) for each point $x\in X$ with ${\mathrm{nw}}_{\chi }\left(x\right)={\aleph }_0$ there is an injective sequence ${\left(x_n\right)}_{n\in \omega }$ in $X$ that $ℱ$-converges to $x$ for some meager filter $ℱ$ on $\omega$; (c) if a functionally Hausdorff space $X$ contains an $ℱ$-convergent injective sequence for some meager filter...

Answering a question of Telgársky in the negative, it is shown that there is a space which is β-favorable in the strong Choquet game, but all of its nonempty $W_{\delta }$-subspaces are of the second category in themselves.

We call a subset S of a topological vector space V linearly Borel if for every finite number n, the set of all linear combinations of S of length n is a Borel subset of V. It is shown that a Hamel basis of an infinite-dimensional Banach space can never be linearly Borel. This answers a question of Anatoliĭ Plichko.

In this note we show the following theorem: “Let $X$ be an almost $k$-discrete space, where $k$ is a regular cardinal. Then $X$ is $k^{+}$-Baire iff it is a $k$-Baire space and every point-$k$ open cover $𝒰$ of $X$ such that $card\left(𝒰\right)\le k$ is locally-$k$ at a dense set of points.” For $k={\aleph }_0$ we obtain a well-known characterization of Baire spaces. The case $k={\aleph }_1$ is also discussed.