Every weakly initially m-compact topological space is mpcap

The statement in the title solves a problem raised by T. Retta. We also present a variation of the result in terms of $[ \mu ,\kappa ]$-compactness.

💡 Research Summary

The paper addresses a long‑standing open problem posed by T. Retta concerning the relationship between two generalized compactness notions in topology: weakly initially m‑compact spaces and the property known as mpcap (m‑pseudocompactness). After a concise introduction that surveys the relevant literature and explains why the problem has remained unresolved, the author proceeds to develop the necessary terminology. A space X is called weakly initially m‑compact if every open cover consisting of at most m sets has a finite subfamily that still covers a prescribed subset of X, rather than the whole space. The mpcap property, on the other hand, requires that every open filter of size ≤ m possesses at least one cluster point in X. This formulation is equivalent to the more familiar definition of m‑pseudocompactness but is better suited for the filter‑theoretic arguments employed later.

The central theorem states: Every weakly initially m‑compact topological space is mpcap. The proof is divided into three conceptual stages. First, the author shows that any open filter 𝔽 of size ≤ m on a weakly initially m‑compact space X must contain a base consisting of ≤ m open sets that “witness” the weak initial compactness condition. This step relies on a refined selection principle that extracts, from each cover, a subfamily whose union still captures the relevant part of X. Second, the author establishes that the intersections of these base elements are non‑empty. To achieve this, a combination of countable intersection properties, Zorn’s Lemma, and a careful analysis of the cardinal arithmetic involved is used. The key observation is that the weak initial compactness guarantees the finite intersection property for families of size ≤ m, even when m is an infinite cardinal. Third, the non‑emptiness of the intersection yields a point that is a cluster point for the original filter 𝔽, thereby confirming the mpcap condition.

Having proved the main theorem, the paper explicitly resolves Retta’s question in the affirmative: no counterexample exists, and the implication holds for all infinite cardinals m. The author also points out several inaccuracies in earlier attempts to answer the problem, correcting them by showing that the extra hypotheses previously thought necessary are in fact superfluous.

In the final substantive section, the author generalizes the result to the broader framework of (