We consider the class of decreasing (G) spaces introduced by Collins and Roscoe and address the question as to whether it coincides with the class of decreasing (A) spaces. We provide a partial solution to this problem (the answer is yes for homogeneous spaces). We also express decreasing (G) as a monotone normality type condition and explore the preservation of decreasing (G) type properties under closed maps. The corresponding results for decreasing (A) spaces are unknown.

We consider definably complete Baire expansions of ordered fields: every definable subset of the domain of the structure has a supremum and the domain cannot be written as the union of a definable increasing family of nowhere dense sets. Every expansion of the real field is definably complete and Baire, and so is every o-minimal expansion of a field. Moreover, unlike the o-minimal case, the structures considered form an axiomatizable class. In this context we prove a version of the Kuratowski-Ulam...

We consider the space $D(X,Y)$ of densely continuous forms introduced by Hammer and McCoy and investigated also by Holá . We show some additional properties of $D(X,Y)$ and investigate the subspace $D^{*}\left(X\right)$ of locally bounded real-valued densely continuous forms equipped with the topology of pointwise convergence ${\tau }_p$. The largest part of the paper is devoted to the study of various cardinal functions for $(D^{*}\left(X\right),{\tau }_p)$, in particular: character, pseudocharacter, weight, density, cellularity, diagonal degree, $\pi$-weight, $\pi$-character,...

We relate some subsets $G$ of the product $X\times Y$ of nonseparable Luzin (e.g., completely metrizable) spaces to subsets $H$ of ${\mathbb{N}}^{\mathbb{N}}\times Y$ in a way which allows to deduce descriptive properties of $G$ from corresponding theorems on $H$. As consequences we prove a nonseparable version of Kondô’s uniformization theorem and results on sets of points $y$ in $Y$ with particular properties of fibres $f^{-1}\left(y\right)$ of a mapping $fX\to Y$. Using these, we get descriptions of bimeasurable mappings between nonseparable Luzin spaces in terms of fibres.

This paper was extensively circulated in manuscript form beginning in the Summer of 1989. It is being published here for the first time in its original form except for minor corrections, updated references and some concluding comments.

For a space X and a regular uncountable cardinal κ ≤ |X| we say that κ ∈ D(X) if for each $T\subset X^2-\Delta \left(X\right)$ with |T| = κ, there is an open neighborhood W of Δ(X) such that |T - W| = κ. If ${\omega }_1\in D\left(X\right)$ then we say that X has a small diagonal, and if every regular uncountable κ ≤ |X| belongs to D(X) then we say that X has an H-diagonal. In this paper we investigate the interplay between D(X) and topological properties of X in the category of generalized ordered spaces. We obtain cardinal invariant theorems and metrization theorems...

Jachymski showed that the set $$\left\{(x,y)\in {𝐜}_0\times {𝐜}_0:{\left(\sum _{i=1}^n\alpha \left(i\right)x\left(i\right)y\left(i\right)\right)}_{n=1}^{\infty }\text{is}\text{bounded}\right\}$$ is either a meager subset of ${𝐜}_0\times {𝐜}_0$ or is equal to ${𝐜}_0\times {𝐜}_0$. In the paper we generalize this result by considering more general spaces than ${𝐜}_0$, namely ${𝐂}_0\left(X\right)$, the space of all continuous functions which vanish at infinity, and ${𝐂}_b\left(X\right)$, the space of all continuous bounded functions. Moreover, we replace the meagerness by $\sigma$-porosity.