×

zbMATH — the first resource for mathematics

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
PDF BibTeX Cite
Full Text: DOI
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
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.