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Spaces hereditarily of \(\kappa\)-type and point \(\kappa\)-type. (English) Zbl 0802.54019
For a space \(X\), and set \(C\subset X\), the character of \(C\) in \(X\) is defined to be the smallest infinite cardinal \(\kappa\) such that there is a family \({\mathcal B}\) of open sets with \(| {\mathcal B}|\leq \kappa\) such that for every open \(U\supset C\) there exists \(B\in {\mathcal B}\) such that \(C\subset B\subset U\). A space \(X\) is said to have point \(\kappa\)- type provided every point in \(X\) is contained in a compact set \(C\) of character at most \(\kappa\). A space is said to have \(\kappa\)-type if every compact set is contained in a compact set of character at most \(\kappa\). Point \(\omega\)-type was introduced by A. V. Arkhangel’skij [Soviet Math., Dokl. 4, 1051-1055 (1968); translation from Dokl. Akad. Nauk SSSR 151, 751-754 (1963; Zbl 0124.158)], and \(\omega\)-type was introduced by M. Henriksen and J. R. Isbell under a different name [Duke Math. J. 25, 83-105 (1958; Zbl 0081.386)]. Among other results, the authors prove that a regular space \(X\) is hereditarily of point \(\kappa\)-type if and only if for every \(p\in X\) there is a set \(E_ p\) (possibly empty) of isolated points of \(X\) such that \(E_ p\cup \{p\}\) is a compact set of character at most \(\kappa\). An analogous result is obtained for regular spaces hereditarily of \(\kappa\)-type.
54D50 \(k\)-spaces
54D30 Compactness
54B05 Subspaces in general topology
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[1] Arhangel’skii A.V.,On a class of spaces containing all metric and locally bicompact spaces, Dokl. Akad. Nauk SSSR151 (1963), 751–754=Soviet Math. Dokl.4 (1968), 1051–1055.
[2] Arhangel’skii A.V.,Bicompact sets and the topology of spaces, Trudy Moskov. Mat. Obsc.13 (1965), 3–55=Trans. Moscow Math. Soc. (1965), 1–62.
[3] Arhangel’skii A.V.,On hereditary properties, Gen. Top. and Appl. (1973), 39–46.
[4] Fletcher P., Lindgren W.,{\(\theta\)}-spaces Gen. Top. and Appl.9 (1957), 139–153. · Zbl 0394.54015
[5] Hneriksen M., Isbell J.,Some properties of compactifications, Duke Math. J.25 (1957), 873–105.
[6] Ismail M.,A note on a theorem of Arhangel’skii, Gen. Top. and Appl.9 (1978), 217–220. · Zbl 0393.54005
[7] Juhasz I.,Cardinal functions in topology, Math. Centre Tracts34, Amsterdam (1971).
[8] Pytkeev E.,Hereditarily plumed spaces, Math. Zametki28 (1980), 603–618. · Zbl 0449.54027
[9] Sabella R.,Spaces in which compact sets have countable local bases, Proc. Amer. Math. Soc.48 (1975), 499–504. · Zbl 0299.54021
[10] Vaughan J.,Spaces of countable and point-countable type, Trans. Amer. Math. Soc.151 (1970), 341–351. · Zbl 0203.55403
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