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

Ultrafilter convergence in ordered topological spaces

We characterize ultrafilter convergence and ultrafilter compactness in linearly ordered and generalized ordered topological spaces. In such spaces, and for every ultrafilter $D$, the notions of $D$-compactness and of $D$-pseudocompactness are equivalent. Any product of initially $\lambda$-compact generalized ordered topological spaces is still initially $\lambda$-compact. On the other hand, preservation under products of certain compactness properties are independent from the usual axioms for set theory.Comment: v. 2: some additions and some improvement

On the J\'onsson distributivity spectrum

Suppose throughout that $\mathcal V$ is a congruence distributive variety. If $m \geq 1$, let $J _{ \mathcal V} (m)$ be the smallest natural number $k$ such that the congruence identity $\alpha ( \beta \circ \gamma \circ \beta \dots ) \subseteq \alpha \beta \circ \alpha \gamma \circ \alpha \beta \circ \dots$ holds in $\mathcal V$, with $m$ occurrences of $\circ$ on the left and $k$ occurrences of $\circ$ on the right. We show that if $J _{ \mathcal V} (m) =k$, then $J _{ \mathcal V} (m \ell ) \leq k \ell$, for every natural number $\ell$. The key to the proof is an identity which, through a variety, is equivalent to the above congruence identity, but involves also reflexive and admissible relations. If $J _{ \mathcal V} (1)=2$, that is, $\mathcal V$ is $3$-distributive, then $J _{ \mathcal V} (m) \leq m$, for every $m \geq 3$ (actually, a more general result is presented which holds even in nondistributive varieties). If $\mathcal V$ is $m$-modular, that is, congruence modularity of $\mathcal V$ is witnessed by $m+1$ Day terms, then $J _{ \mathcal V} (2) \leq J _{ \mathcal V} (1) + 2m^2-2m -1$. Various problems are stated at various places.Comment: v. 4, added somethin

Some more Problems about Orderings of Ultrafilters

We discuss the connection between various orders on the class of all the ultrafilters and certain compactness properties of abstract logics and of topological spaces. We present a model theoretical characterization of Comfort order. We introduce a new order motivated by considerations in abstract model theory. For each of the above orders, we show that if $E$ is a $(\lambda, \lambda)$-regular ultrafilter, and $D$ is not $(\lambda, \lambda)$-regular, then $E \not \leq D$. Many problems are stated.Comment: 7 page