A characterization of the compact spaces with a retractional skeleton
Keywords:
$r$-skeleton, $c$-skeleton, Valdivia compact spaces, Corson compact spacesAbstract
In [14] the authors introduced the concept of retractional skeleton ( $r$-skeleton) and they characterized the Valdivia compact spaces with this notion. In [2], the authors introduced the notion of (full) $c$-skeleton which is a pair consisting of a family of closed subsets and an $\omega$-monotone map which satisfy certain properties; and they proved that a compact space is a Corson compact space iff it admits a full $c$-skeleton. The main purpose of this article is to prove that a compact space admits a retractional skeleton iff it admits a $c$-skeleton. Moreover, we establish a condition on a $c$-skeleton which allows a compact space be a Valdivia compact space. Additionally, we provide an answer to Question 5.8 of the paper [2] by using an argument different from the one used in [11] to respond to the same question.
References
I. Bandlow, A characterization of Corson-compact spaces, Comment. Math. Univ. Carolin., 32, 3 (1991), 545-550.
F. Casarrubias-Segura, S. García-Ferreira and R. Rojas-Hernández, Characterizing Corson and Valdivia compact spaces, Journal of Mathematical Analysis and Applications, 451,2 (2017), 1154-1164.
C. Correa, M. Cúth and J. Somaglia, Characterization of (semi-)Eberlein compacta using retractional skeletons, Studia Math., (2021), online first.
C. Correa, M. Cúth and J. Somaglia, Characterization of weakly K-analytic and Vašák spaces using projectional skeletons and using separable PRI, (2021), preprint available at arxiv.org.
M. Cúth, Noncommutative Valdivia compacta, Commentationes Mathematicae Universitatis Carolinae, 55,1 (2014), 53-72.
M. Cúth and O. Kalenda, Monotone retractability and retractional skeletons, Journal of Mathematical Analysis and Applications, 423 (2015), 18-31.
R. Engelking, General Topology, 2nd ed., Sigma Ser. Pure Math. 6 , Herdermann, Berlin, 1989.
M. J. Fabian, Gáteaux differentiability of covex functions and topology: Weak Asplund spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, J. Wiley & Son. Inc., New York, 1997., A. Wiley-Interscience Publication.
S. García-Ferreira and R. Rojas-Hernández, Families of continuous retractions and function spaces, Journal of Mathematical Analysis and Applications, 441,1 (2016), 330-348.
S. García-Ferreira and C. Yescas-Aparicio, Families of retractions and families of closed subsets on compact spaces, Topology and its Applications, 301 (2021), 107504.
F. Hernández-Hernández and R. Rojas-Hernández, Countably compact spaces admitting full $r$-skeletons are proximal, Topology and its Applications, 299 (2021), 107733.
O. F. K. Kalenda, On the class of continuous images of Valdivia compacta, Extracta Math., 18,1 (2003), 65-80.
W. Kubiś, Banach spaces with projectional skeletons, Journal of Mathematical Analysis and Applications, 350,2 (2009), 758-776.
W. Kubiś and H. Michalewski, Small Valdivia compact spaces, Topology and its Applications, 153,14 (2006), 2560-2573.
R. Rojas-Hernández, Sobre la estructura de los compactos de Corson Topología y sus Aplicaciones 6, Benemérita Universidad Autónoma de Puebla, (2018).
R. Rojas-Hernández, Function spaces and D-property, Topology Proceedings, 43 (2014) 301317.
G. A. Sokolov, On some class of compact spaces lying in $\Sigma$-products, Commentationes Mathematicae Universitatis Carolinae, 25,2 (1984), 219-231.
V. Tkachuk, Lifting the Collins-Roscoe property by condensations, Topology Proceedings, 42 (2013), 1-15.
