Symmetric compactifications of the integers and separable quotients of spaces Cp(X)
Abstract
A compactification of the discrete space ω of all integers is called symmetric if it is the quotient space obtained by gluing together the remainders of two copies of some other compactification of ω. This is a generalization of both a convergent sequence, which is in a way the minimal symmetric compactification of ω, and the Arkhangel'ski\uı--Bereznitski\uı--Schachermayer space studied in Cp-theory, which is in a sense the maximal symmetric compactification of ω. We investigate symmetric compactifications of ω and their relations to the Separable Quotient Problem for spaces Cp(X) and to the existence of Josefson--Nissenzweig sequences of finitely supported Borel measures on spaces X, in particular with supports of bounded size. Further, we reduce the Separable Quotient Problem for spaces Cp(K), K compact, to the case when K is a totally asymmetric compactification of ω. Our results shed some new light on the Grothendieck property of Banach spaces C(K).
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BibTeX
@misc{silber2026symmetric,
title = {{Symmetric compactifications of the integers and separable quotients of spaces $C_p(X)$}},
author = {Zdeněk Silber and Damian Sobota},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18904},
eprint = {2609.18904},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18904},
}
RIS
TY - UNPB
TI - Symmetric compactifications of the integers and separable quotients of spaces C_p(X)
AU - Zdeněk Silber
AU - Damian Sobota
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18904
UR - https://arxiv.org/abs/2609.18904
AB - A compactification of the discrete space ω of all integers is called symmetric if it is the quotient space obtained by gluing together the remainders of two copies of some other compactification of ω. This is a generalization of both a convergent sequence, which is in a way the minimal symmetric compactification of ω, and the Arkhangel'ski\uı--Bereznitski\uı--Schachermayer space studied in C_p-theory, which is in a sense the maximal symmetric compactification of ω. We investigate symmetric compactifications of ω and their relations to the Separable Quotient Problem for spaces C_p(X) and to the existence of Josefson--Nissenzweig sequences of finitely supported Borel measures on spaces X, in particular with supports of bounded size. Further, we reduce the Separable Quotient Problem for spaces C_p(K), K compact, to the case when K is a totally asymmetric compactification of ω. Our results shed some new light on the Grothendieck property of Banach spaces C(K).
ER -