More generalizations of pseudocompactness

We introduce a covering notion depending on two cardinals, which we call $\mathcal O$-$[ \mu, \lambda ]$-compactness, and which encompasses both pseudocompactness and many other generalizations of pseudocompactness. For Tychonoff spaces, pseudocompactness turns out to be equivalent to $\mathcal O$-$[ \omega, \omega ]$-compactness. We provide several characterizations of $\mathcal O$-$[ \mu, \lambda ]$-compactness, and we discuss its connection with $D$-pseudocompactness, for $D$ an ultrafilter. We analyze the behaviour of the above notions with respect to products. Finally, we show that our results hold in a more general framework, in which compactness properties are defined relative to an arbitrary family of subsets of some topological space $X$.Comment: 22 page

Products of sequentially compact spaces and compactness with respect to a set of filters

Let $X$ be a product of topological spaces. $X$ is sequentially compact if and only if all subproducts by $\leq \mathfrak s$ factors are sequentially compact. If $\mathfrak s = \mathfrak h$, then $X$ is sequentially compact if and only if all factors are sequentially compact and all but at most $<\mathfrak s$ factors are ultraconnected. We give a topological proof of the inequality $cf \mathfrak s \geq \mathfrak h$. Recall that $\mathfrak s$ denotes the splitting number, and $\mathfrak h$ the distributivity number. Parallel results are obtained for final $\omega_n$-compactness and for other properties, as well as in the general context of a formerly introduced notion of compactness with respect to a set of filters. Some corresponding invariants are introduced.Comment: v3, entirely rewritten with many additions v4, fixed some detail

Day's Theorem is sharp for nn even

Both congruence distributive and congruence modular varieties admit Maltsev characterizations by means of the existence of a finite but variable number of appropriate terms. A. Day showed that from J\'onsson terms $t_0, \dots, t_n$ witnessing congruence distributivity it is possible to construct terms $u_0, \dots, u _{2n-1}$ witnessing congruence modularity. We show that Day's result about the number of such terms is sharp when $n$ is even. We also deal with other kinds of terms, such as alvin, Gumm, directed, specular, mixed and defective. All the results hold also when restricted to locally finite varieties. We introduce some families of congruence distributive varieties and characterize many congruence identities they satisfy.Comment: v.2, some improvements and some corrections, particularly in Section 9 v.3, a few further improvements, corrections simplification