The category of compactifications and its coreflections

Anthony W. Hager; Brian Wynne

Commentationes Mathematicae Universitatis Carolinae (2022)

  • Volume: 62 63, Issue: 3, page 365-378
  • ISSN: 0010-2628

Abstract

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We define “the category of compactifications”, which is denoted CM, and consider its family of coreflections, denoted corCM. We show that corCM is a complete lattice with bottom the identity and top an interpretation of the Čech–Stone β . A c ∈ corCM implies the assignment to each locally compact, noncompact Y a compactification minimum for membership in the “object-range” of c . We describe the minimum proper compactifications of locally compact, noncompact spaces, show that these generate the atoms in corCM (thus corCM is not a set), show that any c ∈ corCM not the identity is above an atom, and that β is not the supremum of atoms.

How to cite

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Hager, Anthony W., and Wynne, Brian. "The category of compactifications and its coreflections." Commentationes Mathematicae Universitatis Carolinae 62 63.3 (2022): 365-378. <http://eudml.org/doc/299035>.

@article{Hager2022,
abstract = {We define “the category of compactifications”, which is denoted CM, and consider its family of coreflections, denoted corCM. We show that corCM is a complete lattice with bottom the identity and top an interpretation of the Čech–Stone $\beta $. A $c \in $corCM implies the assignment to each locally compact, noncompact $Y$ a compactification minimum for membership in the “object-range” of $c$. We describe the minimum proper compactifications of locally compact, noncompact spaces, show that these generate the atoms in corCM (thus corCM is not a set), show that any $c \in $corCM not the identity is above an atom, and that $\beta $ is not the supremum of atoms.},
author = {Hager, Anthony W., Wynne, Brian},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {compactification; coreflection; atom in a lattice},
language = {eng},
number = {3},
pages = {365-378},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The category of compactifications and its coreflections},
url = {http://eudml.org/doc/299035},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - Hager, Anthony W.
AU - Wynne, Brian
TI - The category of compactifications and its coreflections
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 3
SP - 365
EP - 378
AB - We define “the category of compactifications”, which is denoted CM, and consider its family of coreflections, denoted corCM. We show that corCM is a complete lattice with bottom the identity and top an interpretation of the Čech–Stone $\beta $. A $c \in $corCM implies the assignment to each locally compact, noncompact $Y$ a compactification minimum for membership in the “object-range” of $c$. We describe the minimum proper compactifications of locally compact, noncompact spaces, show that these generate the atoms in corCM (thus corCM is not a set), show that any $c \in $corCM not the identity is above an atom, and that $\beta $ is not the supremum of atoms.
LA - eng
KW - compactification; coreflection; atom in a lattice
UR - http://eudml.org/doc/299035
ER -

References

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  8. Hager A. W., Wynne B., 10.1016/j.topol.2020.107402, Topology Appl. 289 (2021), Paper No. 107402, 9 pages. MR4192355DOI10.1016/j.topol.2020.107402
  9. Herrlich H., Topologische Reflexionen und Coreflexionen, Lecture Notes in Mathematics, 78, Springer, Berlin, 1968 (German). Zbl0182.25302MR0256332
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