Remainders in compactifications and generalized metrizability properties. (English) Zbl 1075.54012

The starting point for this article is the classical result of Henriksen and Isbell that a space \(X\) is of countable type if and only if the remainder in any (or in some) compactification of \(X\) is Lindelöf. This result implies for example that every remainder of a metrizable space is Lindelöf. A space \(X\) has property \(\mathcal P\) at infinity if some remainder of \(X\) has the property \(\mathcal P\). Hence each metrizable space is Lindelöf at infinity. The author considers spaces whose remainders are, in some sense, close to being metrizable and presents interesting results about them. He proves for example that every remainder of a Lindelöf \(p\)-space is a Lindelöf \(p\)-space. This does not generalize to paracompact \(p\)-spaces. Interestingly, for topological groups, it does generalize. Several open problems are posed. One of them is to characterize the nowhere locally compact topological groups that have a metrizable remainder.


54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54B05 Subspaces in general topology
54D40 Remainders in general topology
54H11 Topological groups (topological aspects)
Full Text: DOI


[1] Arhangel’skii, A. V., On a class of spaces containing all metric and all locally compact spaces, Mat. Sb.. Mat. Sb., Amer. Math. Soc. Transl., 92, 109, 1-39 (1970), English translation: · Zbl 0206.24804
[2] Arhangel’skii, A. V., Classes of topological groups, Uspekhi Mat. Nauk, 36, 3, 127-146 (1981) · Zbl 0468.22001
[3] Arhangel’skii, A. V., Topological invariants in algebraic environment, (Hušek, M.; van Mill, J., Recent Progress in General Topology, vol. 2 (2002), North-Holland: North-Holland Amsterdam), 1-57 · Zbl 1030.54026
[4] Arhangelskii, A. V., Some connections between properties of topological groups and of their remainders, Moscow Univ. Math. Bull., 54, 3, 1-6 (1999) · Zbl 0949.54054
[5] Arhangel’skii, A. V.; Bella, A., Cardinal invariants in remainders and variations of tightness, Proc. Amer. Math. Soc., 119, 3, 947-954 (1993) · Zbl 0786.54002
[6] Ceder, J., Some generalizations of metric spaces, Pacific J. Math., 11, 105-126 (1961) · Zbl 0103.39101
[7] Engelking, R., General Topology (1977), PWN: PWN Warszawa · Zbl 0373.54002
[8] Filippov, V. V., On perfect images of paracompact \(p\)-spaces, Soviet Math. Dokl., 176, 533-536 (1967)
[9] Henriksen, M.; Isbell, J. R., Some properties of compactifications, Duke Math. J., 25, 83-106 (1958) · Zbl 0081.38604
[10] Nagami, K., \(Σ\)-spaces, Fund. Math., 61, 169-192 (1969) · Zbl 0181.50701
[11] Roelke, W.; Dierolf, S., Uniform Structures on Topological Groups and Their Quotients (1981), McGraw-Hill: McGraw-Hill New York · Zbl 0489.22001
[12] Shapirovskij, B. E., On separability and metrizability of spaces with Souslin’s condition, Soviet Math. Dokl., 13, 6, 1633-1638 (1972) · Zbl 0268.54007
[13] Tkachenko, M. G., On the Souslin property in free topological groups over compact Hausdorff spaces, Math. Notes, 34 (1983) · Zbl 0535.22002
[14] Uspenskij, V. V., A topological group generated by a Lindelöf \(Σ\)-space has the Souslin property, Soviet Math. Dokl., 26, 166-169 (1982) · Zbl 0527.22001
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