We introduce the properties of a space to be strictly $WFU\left(M\right)$ or strictly $SFU\left(M\right)$, where $\varnothing \ne M\subset {\omega }^{*}$, and we analyze them and other generalizations of $p$-sequentiality ($p\in {\omega }^{*}$) in Function Spaces, such as Kombarov’s weakly and strongly $M$-sequentiality, and Kocinac’s $WFU\left(M\right)$ and $SFU\left(M\right)$-properties. We characterize these in $C_{\pi }\left(X\right)$ in terms of cover-properties in $X$; and we prove that weak $M$-sequentiality is equivalent to $WFU\left(L\right(M\left)\right)$-property, where $L\left(M\right)=\{{}^{\lambda }p:\lambda <{\omega }_1$ and $p\in M\}$, in the class of spaces which are $p$-compact for every $p\in M\subset {\omega }^{*}$; and that $C_{\pi }\left(X\right)$ is a $WFU\left(L\right(M\left)\right)$-space iff...

We introduce and study a natural class of variable exponent ${\ell }^p$ spaces, which generalizes the classical spaces ${\ell }^p$ and c₀. These spaces will typically not be rearrangement-invariant but instead they enjoy a good local control of some geometric properties. Some geometric examples are constructed by using these spaces.

Let $X$ be a locally convex space, $m:\Sigma \to X$ be a vector measure defined on a $\sigma$-algebra $\Sigma$, and $L^1\left(m\right)$ be the associated (locally convex) space of $m$-integrable functions. Let $\Sigma \left(m\right)$ denote $\{{\chi }_{{}_E};E\in \Sigma \}$, equipped with the relative topology from $L^1\left(m\right)$. For a subalgebra $𝒜\subseteq \Sigma$, let ${𝒜}_{\sigma }$ denote the generated $\sigma$-algebra and ${\overline{𝒜}}_s$ denote the closure of $\chi \left(𝒜\right)=\{{\chi }_{{}_E};E\in 𝒜\}$ in $L^1\left(m\right)$. Sets of the form ${\overline{𝒜}}_s$ arise in criteria determining separability of $L^1\left(m\right)$; see [6]. We consider some natural questions concerning ${\overline{𝒜}}_s$ and, in particular, its relation to $\chi \left({𝒜}_{\sigma }\right)$. It is shown that...

We provide a complete isomorphic classification of the Banach spaces of continuous functions on the compact spaces $2^{}\oplus [0,\alpha ]$, the topological sums of Cantor cubes $2^{}$, with smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. In particular, we prove that it is relatively consistent with ZFC that the only isomorphism classes of $C\left(2^{}\oplus [0,\alpha ]\right)$ spaces with ≥ ℵ₀ and α ≥ ω₁ are the trivial ones. This result leads to some elementary questions on large cardinals.

We present an extension of the classical isomorphic classification of the Banach spaces C([0,α]) of all real continuous functions defined on the nondenumerable intervals of ordinals [0,α]. As an application, we establish the isomorphic classification of the Banach spaces $C\left(2^{}\times [0,\alpha ]\right)$ of all real continuous functions defined on the compact spaces $2^{}\times [0,\alpha ]$, the topological product of the Cantor cubes $2^{}$ with smaller than the first sequential cardinal, and intervals of ordinal numbers [0,α]. Consequently,...

It is shown that a space $X$ is $L\left({}^{\mu }p\right)$-Weakly Fréchet-Urysohn for $p\in {\omega }^{*}$ iff it is $L\left({}^{\nu }p\right)$-Weakly Fréchet-Urysohn for arbitrary $\mu ,\nu <{\omega }_1$, where ${}^{\mu }p$ is the $\mu$-th left power of $p$ and $L\left(q\right)=\{{}^{\mu }q:\mu <{\omega }_1\}$ for $q\in {\omega }^{*}$. We also prove that for $p$-compact spaces, $p$-sequentiality and the property of being a $L\left({}^{\nu }p\right)$-Weakly Fréchet-Urysohn space with $\nu <{\omega }_1$, are equivalent; consequently if $X$ is $p$-compact and $\nu <{\omega }_1$, then $X$ is $p$-sequential iff $X$ is ${}^{\nu }p$-sequential (Boldjiev and Malyhin gave, for each $P$-point $p\in {\omega }^{*}$, an example of a compact space $X_p$ which is ${}^{2}p$-Fréchet-Urysohn...

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...