×

zbMATH — the first resource for mathematics

Some generalizations of the concept of a \(p\)-space. (English) Zbl 1229.54036
The authors define generalizations of the notion of a \(p\)-space. We say that a subset \(X\) of a space \(Z\) is \(s\)-embedded if there is a countable family \(\mathcal S\) of open subsets such that \(X\) can be written as the union of a family of subsets, each of which is the intersection of a subfamily of \(\mathcal S\). An \(s\)-space is one which is \(s\)-embedded in one of its compactifications. A partial pluming of \(X\) in \(Z\) is a sequence \(\langle\gamma_n:n\in\omega\rangle\) of families of open sets such that whenever \(x\in X\) and \(z\in Z\setminus X\) there is an \(n\) such that \(x\in\bigcup\gamma_n\) and \(z\notin\text{St}(x,\gamma_n)\); the sequence is a pluming if \(X\subset\bigcup\gamma_n\) for all \(n\). If \(X\) has a (partial) pluming in \(Z\) then it is said to be \(p\)-embedded (\(p^*\)-embedded) in \(Z\); a \(p\)-space (\(p^*\)-space) is one that is \(p\)-embedded (\(p^*\)-embedded) in some compactification.
Besides establishing the basic properties of and the relations between these notions the authors also apply them to sufficient conditions for and characterizations of being a \(p\)-space or metrizable.
Reviewer: K. P. Hart (Delft)

MSC:
54E18 \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc.
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
PDF BibTeX Cite
Full Text: DOI
References:
[1] Alleche, B.; Arhangelʼskii, A.V.; Calbrix, J., Weak developments and metrization, Topology appl., 100, 23-38, (2000) · Zbl 0935.54027
[2] Arhangelʼskii, A.V., A class of spaces which contains all metric and all locally compact spaces, Mat. sb., Amer. math. soc. transl., 92, 1-39, (1970), English transl.:
[3] Arhangelʼskii, A.V., Mappings and spaces, Uspekhi mat. nauk, Russian math. surveys, 21, 4, 115-162, (1966), English transl.: · Zbl 0171.43603
[4] Arhangelʼskii, A.V., External bases of sets lying in bicompacta, Dokl. akad. nauk SSSR, Soviet math. dokl., 1, 573-574, (1960), English transl.: · Zbl 0095.37402
[5] Arhangelʼskii, A.V., The general concept of cleavability of a topological space, Topology appl., 44, 27-36, (1992) · Zbl 0786.54014
[6] Arhangelʼskii, A.V.; Choban, M.M., Completeness type properties of semitopological groups, and the theorems of Montgomery and Ellis, Topology proc., 37, 33-60, (2011), E-published on April 29, 2010 · Zbl 1213.54051
[7] Arhangelʼskii, A.V.; Choban, M.M., Semitopological groups, and the theorems of Montgomery and Ellis, C. R. acad. bulgare sci., 62, 8, 917-922, (2009) · Zbl 1199.54190
[8] Arhangelʼskii, A.V.; Choban, M.M.; Kenderov, P.S., Topological games and continuity of group operations, Topology appl., 157, 2542-2552, (2010) · Zbl 1206.54031
[9] Arhangelʼskii, A.V.; Tkachenko, M.G., Topological groups and related structures, (2008), Atlantis Press Amsterdam, Paris
[10] Arhangelʼskii, A.V.; Reznichenko, E.A., Paratopological and semitopological groups versus topological groups, Topology appl., 151, 107-119, (2005) · Zbl 1077.54023
[11] Arhangelʼskii, A.V.; Shakhmatov, D.B., On pointwise approximation of arbitrary functions by countable families of continuous functions, Tr. semin. im. I.G. petrovskogo, J. soviet math., 50, 1497-1512, (1990), English transl.: · Zbl 0701.41026
[12] Balogh, Z., On the metrizability of \(F_{p p}\)-spaces and its relationship to the normal Moore space conjecture, Fund. math., 113, 45-58, (1981) · Zbl 0472.54017
[13] Balogh, Z.; Burke, D.K., Two results on spaces with a sharp base, Topology appl., 154, 1281-1285, (2007) · Zbl 1114.54018
[14] Chaber, J.; Čoban, M.M.; Nagami, K., On monotonic generalizations of Moore spaces, čech complete spaces and p-spaces, Fund. math., 84, 107-119, (1974) · Zbl 0292.54038
[15] Choban, M., The open mappings and spaces, Rend. circ. mat. Palermo (2) suppl., 29, 51-104, (1992) · Zbl 0792.54011
[16] Choban, M., Baire sets in complete topological spaces, Ukrainskii mat. J., Ukrainian math. J., 22, 286-295, (1979), English transl.:
[17] Choban, M., Descriptive set theory and topology, (), 157-219, English transl.:
[18] Henriksen, M.; Isbel, J.R., Some properties of compactifications, Duke math. J., 25, 83-106, (1958)
[19] Engelking, R., General topology, (1977), PWN Warszawa
[20] Kenderov, P.S.; Kortezov, I.S.; Moors, W.B., Topological games and topological groups, Topology appl., 109, 157-165, (2001) · Zbl 0976.22003
[21] Kenderov, P.S.; Moors, W.B., Fragmentability and sigma-fragmentability of Banach spaces, J. London math. soc., 60, 203-223, (1999) · Zbl 0953.46004
[22] Kuratowski, K., Topology, vol. 1, (1966), Academic Press New York · Zbl 0158.40901
[23] Morita, K., A survey of the theory of M-spaces, Gen. topol. appl., 1, 49-55, (1971) · Zbl 0213.24002
[24] Pytkeev, E.G., Hereditary plumed spaces, Mat. zametki, Math. notes, 28, 761-769, (1980), English transl.: · Zbl 0462.54018
[25] Wicke, H.H., Open continuous images of certain kinds of M-spaces and completeness of mappings and spaces, Gen. topol. appl., 1, 85-100, (1971) · Zbl 0212.27203
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.