×

zbMATH — the first resource for mathematics

Internal characterization of fragmentable spaces. (English) Zbl 0645.46017
Let X be a topological space and let \(\rho\) be a metric on \(X\times X\). X is said to be fragmented by the metric \(\rho\) if for every \(\epsilon >0\) and each nonempty subset Y of X there is a nonempty relatively open subset U of Y such that \(\rho\)-diam(U)\(\leq \epsilon\) [J. E. Jayne and C. A. Rogers, Acta Math. 155, 41-79 (1985; Zbl 0588.54020)]. The author shows that X is fragmentable if and only if X admits a separating \(\sigma\)-relatively open partitioning. This characterisation is used to prove the main result (Theorem 3.1): Let X be a Hausdorff compact space which is fragmented by a metric. Then \(C(X)^*\) endowed with the weak star topology is a framentable space. Consequently (Corollary 3.6) C(X) is a weak Asplund space.
Reviewer: R.Cross

MSC:
46B20 Geometry and structure of normed linear spaces
54E35 Metric spaces, metrizability
46B25 Classical Banach spaces in the general theory
Citations:
Zbl 0588.54020
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Debs, Mathematika 32 pp 218– (1985)
[2] Coban, Acad. Bulgare Sci. 38 pp 1603– (1985)
[3] Christensen, Math. Scand. 54 pp 70– (1984) · Zbl 0557.46016
[4] Stegall, Mathematika 34 pp 101– (1987)
[5] Sokolov, Comment. Math. Univ. Carolinae 25 pp 219– (1984)
[6] Namioka, Mathematika 34 pp 258– (1987)
[7] DOI: 10.1007/BF02392537 · Zbl 0588.54020
[8] Kenderov, C. R. Acad. Bulgare Sci. 37 (1984)
[9] Kenderov, Acad. Bulgare Sci. 40 pp 17– (1987)
[10] Kenderov, Studia Math. 56 pp 199– (1976)
[11] Hansell, J. fur reine und angew. Math. 361 pp 201– (1985)
[12] DOI: 10.2307/2045974 · Zbl 0622.54020
[13] Leiderman, Mat. Zametki 38 pp 440– (1985)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.