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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.
##### MSC:
 54D50 $$k$$-spaces 54D30 Compactness 54B05 Subspaces in general topology
##### Keywords:
character; point $$\kappa$$-type
Full Text:
##### References:
 [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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