×

Mapping theorems on \(\aleph\)-spaces. (English) Zbl 0663.54017

The following results are proved: Theorem: \(\aleph\)-spaces (i.e. regular spaces with a \(\sigma\)-locally finite \(\kappa\)-network) are pressed by closed Lindelöf continuous maps (f: \(X\to Y\) is Lindelöf if \(f^{- 1}(y)\) is Lindelöf for each \(y\in Y)\). Theorem. A perfect inverse image of an \(\aleph\)-space is an \(\aleph\)-space iff it has a \(G_{\delta}\)- diagonal or a point-countable \(\kappa\)-network.
Reviewer: C.R.Borges

MSC:

54E18 \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc.
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54D55 Sequential spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Foged, L., Characterization of ℵ-spaces, Pacific J. Math., 110, 59-63 (1984) · Zbl 0542.54030
[2] Foged, L., A characterization of closed images of metric spaces, Proc. Amer. Math. Soc., 95, 487-490 (1985) · Zbl 0592.54027
[3] Gittings, R. F., Open mapping theory, (Set-Theoretic Topology (1977), Academic Press: Academic Press New York), 141-191
[4] Gruenhage, G., Generalized metric spaces, (Handbook of Set-Theoretic Topology (1984), North-Holland: North-Holland Amsterdam), 432-502
[5] Gruenhage, G.; Michael, E.; Tanaka, Y., Spaces determined by point-countable covers, Pacific J. Math., 113, 303-332 (1984) · Zbl 0561.54016
[6] Michael, E., ℵ-spaces, J. Math. Mech., 15, 983-1002 (1966) · Zbl 0148.16701
[7] O’Meara, P., A new class of topological spaces, University of Alberta Dissertation (1966)
[8] Tanaka, Y., A characterization for the products of \(k\)-and-ℵ-spaces and related results, Proc. Amer. Math. Soc., 59, 149-155 (1976) · Zbl 0336.54026
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.