We show that, under suitably general formulations, covering properties, accumulation properties and filter convergence are all equivalent notions. This general correspondence is exemplified in the study of products. We prove that a product is Lindelöf if and only if all subproducts by $\le {\omega }_1$ factors are Lindelöf. Parallel results are obtained for final ${\omega }_n$-compactness, $[\lambda ,\mu ]$-compactness, the Menger and the Rothberger properties.

Generalizations of earlier negative results on Property $\left(a\right)$ are proved and two questions on an $\left(a\right)$-version of Jones’ Lemma are posed. We discuss these questions in the realm of locally compact spaces. Using dominating families of functions as a tool, we prove that under the assumptions “$2^{\omega }$ is regular” and “$2^{\omega }<2^{{\omega }_1}$” the existence of a $T_1$ separable locally compact $\left(a\right)$-space with an uncountable closed discrete subset implies the existence of inner models with measurable cardinals. We also use cardinal invariants...

A characterization of the cozero elements of a frame, without reference to the reals, is given and is used to obtain a characterization of pseudocompactness also independent of the reals. Applications are made to the congruence frame of a $\sigma$-frame and to Alexandroff spaces.

Given a Tychonoff space $X$, a base $\alpha$ for an ideal on $X$ is called pseudouniform if any sequence of real-valued continuous functions which converges in the topology of uniform convergence on $\alpha$ converges uniformly to the same limit. This paper focuses on pseudouniform bases for ideals with particular emphasis on the ideal of compact subsets and the ideal of all countable subsets of the ground space.