bibkey: frolik1960countablycompact
authors: Zdeněk Frolík
year: 1960
title: The topological product of countably compact spaces
doi: 10.21136/cmj.1960.100417
url: https://dml.cz/handle/10338.dmlcz/100417
claim: Lemmas 2.8 and 2.9 give classical accumulation-point abundance, cardinal avoidance and omega1 closing-off for separable metrizable spaces in their Stone-Cech compactifications.
strata_touched:
  - D5/S3/Topology/CountablyCompact/InjectiveSelectorClosure
  - D5/S3/Topology/StoneCech/FreeUltrafilterCardinality
license: citation-only
triage: anchor

Classical abundance, avoidance and closing-off

The article appears in Czechoslovak Mathematical Journal 10 (85) (1960), no. 3, 329–338. The directly inspected primary passages are Lemma 2.8 and its proof, printed pages 333–334 (PDF pages 6–7), and Lemma 2.9 and its proof, printed page 334 (PDF page 7). The reading uses the scanned page images; the PDF text extraction contains no article text and is not a transcription of these passages. This note supplies citations and paraphrases, without reproducing the publisher PDF.

Write $\mathfrak{c}=2^{{\aleph }_0}$ and let $\beta P$ denote the Stone–Čech compactification of $P$. Lemma 2.8 states that, for separable metrizable $P$, every infinite $M\subseteq \beta P$ having no accumulation point in $P$ has $2^{\mathfrak{c}}$ accumulation points in $\beta P$. The no-accumulation-in-$P$ hypothesis is essential to this attribution; the lemma is not an unqualified abundance assertion for every infinite subset. It also states that a compact subspace of $\beta P\setminus P$ is either finite or has cardinal $2^{\mathfrak{c}}$.

The proof takes a countably infinite discrete $X\subseteq M$ and uses normality of $P\cup X$ to extend bounded continuous functions from $X$. It identifies the closure of $X$ in $\beta P$ with $\beta X$ and, on printed page 334, explicitly invokes Pospíšil’s theorem that the Stone–Čech compactification of a countably infinite discrete space has cardinal $2^{\mathfrak{c}}$. The footnote on printed page 333 credits M. Katětov with communicating this simple proof. The Pospíšil and Katětov credits are attested by Frolík; their original sources were not independently read here.

Lemma 2.9 states that, for separable metrizable $P$ and $M\subseteq \beta P\setminus P$ with $|M|<2^{\mathfrak{c}}$, there is a countably compact subspace $R$ satisfying $P\subseteq R\subseteq \beta P\setminus M$ and $|R|\le \mathfrak{c}$. Its proof selects accumulation points outside $M$ and iterates through the countable ordinals. A countably infinite subset of the final union lies in a bounded stage, and a later stage contains its selected accumulation point. The cardinal estimate uses ${\mathfrak{c}}^{{\aleph }_0}=\mathfrak{c}$ and ${\aleph }_1\le \mathfrak{c}$. This is primary provenance for the classical abundance, cardinal avoidance and omega1 closing-off methods, not new project methods.

These lemmas differ from the repository’s exact arbitrary T1 closure theorem with an arbitrary map $q$, a good-set predicate, an admissible seed and preservation of $q$-injectivity. They do not state that theorem or supply pairwise distinct projected accumulation values. The fixed-free-ultrafilter D-compact construction is separately attested in ginsburgsaks1975ultrafilters; it is not the arbitrary-$q$ theorem either. The two inspected primary pages do not state the Feldman–Wilce finite-block graph endpoint or resolve their Question 1. This is a passage-level comparison, with no claim about absence elsewhere in the literature, novelty or priority.

The FreeUltrafilterCardinality module supplies the lower bound $2^{\mathfrak c}\leq|{u\in\operatorname{Ultrafilter}(\mathbb N): u\text{ contains every cofinite set}}|$. Its classical cardinal-supply attribution is the Pospíšil theorem explicitly invoked by Frolík on printed page 334. The module constructs an independent family from finite traces, extends each Boolean assignment together with the cofinite filter to an ultrafilter, and recovers the assignment by membership. This links the cardinal-supply attribution only; Frolík’s inspected passage is not asserted to contain this exact finite-trace implementation or to supply the later section-good selector or pairwise distinct projected cluster values. No new set-theoretic novelty is claimed.

Verified locator

DOI: 10.21136/cmj.1960.100417. URL: https://dml.cz/handle/10338.dmlcz/100417. Primary DOI: https://doi.org/10.21136/cmj.1960.100417. Stable primary handle: https://dml.cz/handle/10338.dmlcz/100417. The consulted passages are Lemma 2.8, printed pages 333–334 (PDF pages 6–7), and Lemma 2.9, printed page 334 (PDF page 7).