zbMATH — the first resource for mathematics

Analytic spaces and paracompactness of X 2\(\backslash \Delta\). (English) Zbl 0636.54025
The first author [ibid. 17, 287-304 (1984; Zbl 0547.54016)] proved that every compact Hausdorff space X with X \(2\setminus \Delta\) paracompact, where \(\Delta\) is the diagonal in \(X\times X\), is metrizable. The main theorem of the authors is that every regular \(\Sigma\)-space X such that X \(2\setminus \Delta\) is paracompact has a \(G_{\delta}\)-diagonal (hence X is a \(\sigma\)-space). Here \(\Sigma\)-space and \(\sigma\)-space are used in the sense of K. Nagami [Fundam. Math. 65, 169-192 (1969; Zbl 0181.507)]. A corollary to this is that every completely regular p-space X (in the sense of Arkhangel’skij) such that X \(2\setminus \Delta\) is paracompact is metrizable. In Czech. Math. J. 35(110), 43-51 (1985; Zbl 0583.54023), Z. Frolík asked whether every analytic space X with X \(2\setminus \Delta\) paracompact is point-analytic. By showing that Souslin-F subsets of paracompact p-spaces are strong \(\Sigma\)-spaces the authors answer Frolík’s question in the affirmative.
Reviewer: H.H.Wicke

54E18 \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc.
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
Full Text: DOI
[1] Borges, C.J.R., On stratifiable spaces, Pac. J. math., 17, 1-16, (1966) · Zbl 0175.19802
[2] Chaber, J.; Coban, M.; Nagami, K., On monotonic generalizations of Moore spaces, Czech complete spaces, and p-spaces, Fund. math., 83, 107-119, (1974) · Zbl 0292.54038
[3] Fremlin, D., K-analytic spaces with metrizable compacta, Mathematika, 24, 257-261, (1977) · Zbl 0366.54020
[4] Frolik, Z., Reduction of Baire measurability to uniform continuity, Czech. math. J., 35, 110, 43-51, (1985) · Zbl 0583.54023
[5] Frolík, Z.; Holický, P., Analytic and luzin spaces (non-separable case), Top. appl., 19, 129-156, (1985) · Zbl 0579.54026
[6] Frolík, Z.; Holický, P., Decomposability of completely Suslin additive families, Proc. A.M.S., 82, 653-667, (1981) · Zbl 0496.28003
[7] Gruenhage, G., Covering properties on X2δ, W-sets, and compact subsets of σ-products, Top. appl., 17, 287-304, (1984) · Zbl 0547.54016
[8] Gruenhage, G., Generalized metric spaces, () · Zbl 0478.54018
[9] Hansell, R.W.; Jayne, J.E.; Rogers, C.A., K-analytic sets, Mathematika, 30, 189-227, (1983) · Zbl 0524.54028
[10] Michael, E., On Nagami’s σ-spaces and related matters, Proc. Washington state univ. topology conf., 13-19, (1970)
[11] Nagami, K., σ-spaces, Fund. math., 65, 169-192, (1969) · Zbl 0181.50701
[12] Okuyama, A., On metrizability of M-spaces, Proc. Japan acad., 40, 176-179, (1964) · Zbl 0127.38702
[13] Shiraki, T., M-spaces, their generalizations and metrization theorems, Sci. rep. Tokyo kyoiku daigaku, 11, 57-67, (1971), Section A · Zbl 0233.54016
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.