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Classes defined by stars and neighbourhood assignments. (English) Zbl 1131.54022
A neighbourhood assignment in a space \(X\) is a family \(\{O_x:\;x\in X\}\) such that \(x\in O_x\in \tau (X)\) for any \(x\in X\). Given a topological property \(\mathcal P\), the class \({\mathcal P}^*\) dual to \(\mathcal P\) (with respect to neighbourhood assignments) consists of spaces \(X\) such that for any neighbourhood assignment \(\{O_x:\;x\in X\}\) there is \(Y\subseteq X\) with \(Y\in \mathcal P\) and \(\bigcup\{O_x:\;x\in Y\}=X\). The authors show that compactness, \(\kappa\)-compactness, pseudocompactness and linear Lindelöfness are self-dual in this sense. Given a family \(\mathcal U\) of subsets of \(X\), the star, \(St(A,{\mathcal U})\), of the set \(A\) with respect to \(\mathcal U\) is the set \(\bigcup \{U\in{\mathcal U}:A\cap U\neq\emptyset\}\). \(X\) is said to be star-\(\mathcal P\) (or star determined by \(\mathcal P\)), if for any open cover \(\mathcal U\) of the space \(X\) there is a subspace \(Y\subseteq X\), with \(St(Y,{\mathcal U})=X\) and \(Y\in \mathcal P\). The authors show that the classes of pseudocompact spaces, the spaces star determined by countably compact spaces, star compact spaces, and countably compact spaces are all distinct. Among other things, an example is given of (1) A pseudocompact first countable space which is not star determined by countably compact spaces. (2) A space \(X\) which is star determined by metrizable compact spaces but not compact countable spaces. Several problems are left unsolved, such as: Must every star compact topological group be countably compact?

54H11 Topological groups (topological aspects)
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
22A05 Structure of general topological groups
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D25 “\(P\)-minimal” and “\(P\)-closed” spaces
54C25 Embedding
Full Text: DOI
[1] Arhangel’skiıˇ, A.V.; Buziakova, R.Z., Addition theorems and D-spaces, Comment. math. univ. carolinae, 43, 4, 653-663, (2002) · Zbl 1090.54017
[2] Bonanzinga, M.; Matveev, M.V., Centered lindelöfness versus star-lindelöfness, Comment. math. univ. carolinae, 41, 1, 111-122, (2000) · Zbl 1037.54502
[3] Buziakova, R.Z., On D-property of strong σ-spaces, Comment. math. univ. carolinae, 43, 3, 493-495, (2002) · Zbl 1090.54018
[4] Buziakova, R.Z., Hereditary D-property of function spaces over compacta, Proc. amer. math. soc., 132, 11, 3433-3439, (2004) · Zbl 1064.54029
[5] Borges, C.R.; Wehrly, A.C., A study of D-spaces, Topology proc., 16, 7-15, (1991) · Zbl 0787.54023
[6] van Douwen, E.K., Pixley – roy topology on spaces of subsets, (), 111-134 · Zbl 0372.54006
[7] van Douwen, E.K.; Reed, G.M.; Roscoe, A.W.; Tree, I.J., Star covering properties, Topology appl., 39, 71-103, (1991) · Zbl 0743.54007
[8] Engelking, R., General topology, (1977), PWN Warszawa
[9] Fleischman, R., A new extension of countable compactness, Fund. math., 67, 1-9, (1970) · Zbl 0194.54601
[10] Fleissner, W.G.; Stanley, A.M., D-spaces, Topology appl., 114, 261-271, (2001) · Zbl 0983.54024
[11] Ikenaga, S., Some properties of \(w - n\)-star spaces, Research reports of Nara national college of technology, 23, 52-57, (1987)
[12] Ikenaga, S., Topological concept between Lindelöf and pseudo-Lindelöf, Research reports of Nara national college of technology, 26, 103-108, (1990)
[13] Ikenaga, S.; Tani, T., On a topological concept between countable compactness and pseudocompactness, Research reports of numazu technical college, 15, 139-141, (1980)
[14] Hiremath, G.R., On star with Lindelöf center property, J. Indian math. soc., 59, 227-242, (1993) · Zbl 0887.54021
[15] M. Matveev, A survey on star covering properties, Topology Atlas, April 15, 1998
[16] Matveev, M.; Uspenskij, V., On star-compact spaces with a \(G_\delta\)-diagonal, Zb. rad. fil. fac. u nišu, ser. mat., 6, 2, 281-290, (1992) · Zbl 0911.54020
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