arXiv subjects
Publications and source records attributed to Pourya Memarpanahi.
The question whether there is a Lindelof Q-set space or Lindelof $Δ$-set space is considered. We show that J. Moore's ZFC $L$-space is not a Q-set space in ZFC and, assuming all Aronszajn trees are special, it is not a $Δ$-set space.
We define several notions of a limit point on sequences with domain a barrier in $[ω]^{<ω}$ focusing on the two dimensional case $[ω]^2$. 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 $α$-countable compactness for $α\leqω_1$. 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 $βω$ 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.
We introduce a generalization of sequential compactness using barriers on $ω$ extending naturally the notion introduced in [W. Kubiś 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{á}n and C. L{ó}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 Katětov-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.