Small Valdivia compact spaces. (English) Zbl 1138.54024

Kubiś, Wiesław; Michalewski, Henryk

Topology Appl. 153, No. 14, 2560-2573 (2006).

It is shown that the class of Valdivia compact spaces is closed under limits of inverse systems \(\{X_\alpha, r_\alpha^\beta\}_{\alpha,\beta\in\kappa}\) such that \(X_0\) is Valdivia compact and all \(r_\alpha^{\alpha+1}\) are simple retractions. That result especially applies to metric compact spaces \(X_\alpha\) and retractions \(r_\alpha^{\alpha+1}\).
As a corollary, a compact space \(X\) with \(w(X)\leq\omega_1\) is Valdivia iff it is a limit of an inverse system of metric spaces with retractions as bonding maps. Another consequence gives a preservation of Valdivia compacts by retractions onto spaces with \(w(X)\leq\omega_1\) or by open surjections onto zero-dimensional spaces with \(w(X)\leq\omega_1\).
Using functorial properties of the construction of certain inverse systems, the authors prove that the class of Valdivia compacts is stable under continuous weight preserving covariant functors in compact spaces.

Reviewer: Miroslav Hušek (Praha)

MSC:

54D30	Compactness
54B35	Spectra in general topology
54B30	Categorical methods in general topology

Keywords:

Valdivia compact; inverse system; retraction; functors in compact spaces

References:

[1]	Argyros, S.; Mercourakis, S.; Negrepontis, S., Functional-analytic properties of Corson-compact spaces, Studia Math., 89, 3, 197-229 (1988) · Zbl 0656.46014
[2]	Corson, H. H., Normality in subsets of product spaces, Amer. J. Math., 81, 785-796 (1959) · Zbl 0095.37302
[3]	Deville, R.; Godefroy, G., Some applications of projective resolutions of identity, Proc. London Math. Soc. (3), 67, 1, 183-199 (1993) · Zbl 0798.46008
[4]	Dow, A., An introduction to applications of elementary submodels to topology, Topology Proc., 13, 1, 17-72 (1988) · Zbl 0696.03024
[5]	Glicksberg, I., Stone-Čech compactifications of products, Trans. Amer. Math. Soc., 90, 369-382 (1959) · Zbl 0089.38702
[6]	Haydon, R., On a problem of Pełczyński: Milutin spaces, Dugundji spaces and AE(0-dim), Studia Math., 52, 23-31 (1974) · Zbl 0294.46016
[7]	Kalenda, O., Valdivia compact spaces in topology and Banach space theory, Extracta Math., 15, 1, 1-85 (2000) · Zbl 0983.46021
[8]	Kalenda, O., Continuous images and other topological properties of Valdivia compacta, Fund. Math., 162, 2, 181-192 (1999) · Zbl 0989.54019
[9]	Kalenda, O., A characterization of Valdivia compact spaces, Collect. Math., 51, 1, 59-81 (2000) · Zbl 0949.46004
[10]	Kalenda, O., On the class of continuous images of Valdivia compacta, Extracta Math., 18, 1, 65-80 (2003) · Zbl 1159.46306
[11]	Kubiś, W., A representation of retracts of cubes
[12]	Kubiś, W.; Leiderman, A., Semi-Eberlein compact spaces, Topology Proc., 28, 2, 603-616 (2004) · Zbl 1090.54021
[13]	Kubiś, W.; Uspenskij, V., A compact group which is not Valdivia compact, Proc. Amer. Math. Soc., 133, 8, 2483-2487 (2005) · Zbl 1066.54027
[14]	Shapiro, L. B., On \(κ\)-metrizable compact Hausdorff spaces, Dokl. Akad. Nauk. Dokl. Akad. Nauk, Russian Acad. Sci. Dokl. Math., 48, 3, 534-538 (1994), (in Russian); translation in
[15]	Ščepin, E. V., Topology of limit spaces with uncountable inverse spectra, Uspehi Mat. Nauk. Uspehi Mat. Nauk, Russian Math. Surveys, 31, 5, 155-191 (1976), (in Russian); English translation · Zbl 0356.54026
[16]	Valdivia, M., Projective resolution of identity in \(C(K)\) spaces, Arch. Math. (Basel), 54, 5, 493-498 (1990) · Zbl 0707.46009
[17]	Valdivia, M., Simultaneous resolutions of the identity operator in normed spaces, Collect. Math., 42, 3, 265-284 (1991) · Zbl 0788.47024