Sequentially compact separable spaces
We consider the following variation of the Scarborough-Stone problem: Is $X^\kappa$ always countably compact whenever $X$ is separable and sequentially compact?
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arXiv subjects
Cesar Corral
Publications and source records attributed to Cesar Corral.
High dimensional countable compactness and ultrafilters
We define several notions of a limit point on sequences with domain a barrier in $[\omega]^{<\omega}$ focusing on the two dimensional case $[\omega]^2$. By exploring some natural candidates, we show that countable compactness has a number of generalizations in terms of limits of high dimensional sequences and define a particular notion of $\alpha$-countable compactness for $\alpha\leq\omega_1$. We then focus on dimension 2 and compare 2-countable compactness with notions previously studied in the literature. We present a number of counterexamples showing that these classes are different. In particular assuming the existence of a Ramsey ultrafilter, a subspace of $\beta\omega$ which is doubly countably compact whose square is not countably compact, answering a question of T. Banakh, S. Dimitrova and O. Gutik. The analysis of this construction leads to some possibly new types of ultrafilters related to discrete, P-points and Ramsey ultrafilters.
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Infinite dimensional sequential compactness: Sequential compactness based on barriers
We introduce a generalization of sequential compactness using barriers on $\omega$ extending naturally the notion introduced in [W. Kubi\'{s} and P. Szeptycki, On a topological Ramsey theorem, \emph{Canad. Math. Bull.}, 66 (2023), {156}--{165}]. We improve results from [C. Corral and O. Guzm{\'a}n and C. L{\'o}pez-Callejas, High dimensional sequential compactness, \emph{Fund. Math.}] by building spaces that are $\mathcal{B}$-sequentially compact but no $\mathcal{C}$-sequentially compact when the barriers $\mathcal{B}$ and $\mathcal{C}$ satisfy certain rank assumption which turns out to be equivalent to a Kat\v{e}tov-order assumption. Such examples are constructed under the assumption $\mathfrak{b} =\mathfrak{c}$. We also exhibit some classes of spaces that are $\mathcal{B}$-sequentially compact for every barrier $\mathcal{B}$, including some classical classes of compact spaces from functional analysis, and as a byproduct we obtain some results on angelic spaces. Finally we introduce and compute some cardinal invariants naturally associated to barriers.
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New examples of MAD families with pseudocompact hyperspaces
We show that both $\mathfrak{ap=c}$ and $\diamondsuit(\mathfrak{b})$ imply the existence of MAD families with pseudocompact Vietoris hyperspace, substantially expanding the list of models where their existence is known. We also discuss some properties on the structure of fin-intersecting MAD families by providing some new examples of non-fin-intersecting MAD families with special properties. We show that the Baire number of $\omega^*$ is strictly larger than $\mathfrak c$ if and only if for every MAD family $\mathcal A$, $\Psi(\mathcal A)$ and its Vietoris hyperspace are $p$-pseudocompact for some free ultrafilter $p$.
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Fin-intersecting MAD families
We introduce a new class of almost disjoint families which we call fin-intersecting almost disjoint families. They are related to almost disjoint families whose Vietoris Hyperspace of their Isbell-Mr\'owka spaces are pseudocompact. We show that under $\mathfrak p=\mathfrak c$ fin-intersecting MAD families exist generically and they also exist if $\mathfrak{a<s}$, but that there are also non fin-intersecting MAD families in ZFC. We also show that under CH, there exists fin intersecting MAD families which remain like that after adding an arbitrary quantity of Cohen reals and Random reals. These results give more models in which pseudocompact MAD families exist.
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