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Wednesday, 4 November 2026: 4 new papers, 0 cross-lists and 2 replacements in General Topology (math.GN), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We answer four questions on irreducible determined $T_0$-topologies and a related question on monotone determined topologies posed by Xu, Yang and Luo (Topology Appl.\ 382 (2026), 109762). For every $T_0$ topology $\tau$, the family $\tau_r$ used to define its irreducible-derived topology is already a topology and preserves all nonempty irreducible subsets and their closures. Consequently, the least irreducible determined refinement is obtained in one step: $\tau_r=\rd(\tau)=\Rd(\tau)$. For an arbitrary poset $P$, we describe the weakest order-compatible irreducible determined topology $\rho(P)$ using Dedekind--MacNeille cuts and a finite-cover condition involving principal lower sets. The Scott space of the binary-prefix domain provides a common counterexample to saturated heredity for irreducible determined spaces and monotone determined spaces: its maximal-point subspace is Cantor space. We also describe $\rho(P)$-closure by iterating an irreducible-cut operator, characterize Scott continuity of this operator, and give an example requiring two applications of the cut operator. A finite-decomposition condition extends the topology and closure-preservation results to general subset systems. For the Rudin and well-filtered determined systems, we exhibit a common $T_1$ example whose first and second derived topologies are the co-countable and discrete topologies, respectively. Thus the first derived topology need not be determined by the same subset system.
A space is feebly compact if every locally finite family of nonempty open subsets of it is finite; a Tychonoff space is feebly compact if and only if it is pseudocompact. M. Tkachenko asked whether a regular feebly compact semitopological group is a topological group provided that it is separable, has countable tightness, or is a $k$-space; for Tychonoff groups the answer is known to be positive. We answer this question in the separable case: every regular separable feebly compact semitopological group is a topological group. The proof is based on the following result: if $X$ is a separable Baire space, $Y$ is a feebly compact space and $Z$ is a regular space, then every separately continuous map $X\times Y\to Z$ is strongly quasicontinuous. Unlike the known approaches for Tychonoff spaces, the proof does not use real-valued functions. More generally, every regular separable semitopological group which is $\Gamma^{BM}_o$-nonmeager, a Baire-type property defined by a modification of the Banach-Mazur game, is a topological group. This class of groups contains all regular separable Baire groups which are feebly $p$-spaces; feebly $p$-spaces form a wider class of spaces, which contains all feebly compact spaces and all $p$-spaces. The results remain valid for spaces satisfying the assumptions locally. In particular, every regular locally separable locally feebly compact semitopological group is a topological group.
A space is ESLC if it contains neither an $S$-space nor an $L$-space, and NFS is the class of spaces containing no free sequences of length $\omega_1$. We prove that under $\mathrm{MA}(\omega_1)$, if all finite powers of $X$ belong to NFS, then $C_p(X)^\omega$ is ESLC; thus, for subspaces of $C_p(X)$, hereditary Lindel\"ofness, hereditary separability and countable spread coincide. This is derived from a dual form of a theorem of Todor\v{c}evi\'c. A Lindel\"of $\Sigma$-space belongs to NFS if and only if its compact subspaces have countable tightness. Hence $C_p(X)$ contains no $L$-spaces if $X$ is a Lindel\"of $\Sigma$-space of countable tightness and $\mathrm{MA}(\omega_1)$ holds, or if $X$ is any Lindel\"of $\Sigma$-space and PFA holds. Under $\mathrm{MA}(\omega_1)$, all iterated function spaces of Corson compact spaces are ESLC, and in ZFC the space $C_p(X,\{0,1\})$ is ESLC for every zero-dimensional Corson compact space $X$.
A $T_0$-space is called $\omega$-sober if every irreducible closed subset having a countable dense subset is the closure of a unique point. This is precisely $H$-sobriety for the countable irreducible subset system $\mathcal R^\omega$. We give two natural Hofmann--Mislove representations in the general $H$-sobriety framework: filters induced by $H$-sets of $X$ characterize $H$-sobriety, whereas filters induced by $H$-sets of the Smyth power space characterize super $H$-sobriety. Their relation is expressed by property $Q$. A minimal closed set argument with a countable selection proves that $\mathcal R^\omega$ has property $Q$. Consequently, $X$ is $\omega$-sober if and only if its Smyth power space is $\omega$-sober, equivalently, every countable irreducible compact-family filter has a compact saturated representation. We also obtain equivalent recovery, compact-intersection and representation conditions for an enlarged class containing the proper Scott-open filters with countable open bases and the filters induced by countable irreducible subsets. A further equivalent formulation uses traces on countable subspaces. The filtered-family specializations recover well-filteredness and $\omega$-well-filteredness. Explicit examples show that $\omega$-sobriety strictly implies $\omega$-well-filteredness, which strictly implies representation for countable open bases alone.