Lindelöf property and absolute embeddings

Bella, A.; Yaschenko, I.

doi:10.1090/S0002-9939-99-04568-2

Lindelöf property and absolute embeddings
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by A. Bella and I. V. Yaschenko PDF

Proc. Amer. Math. Soc. 127 (1999), 907-913 Request permission

Abstract:

It is proved that a Tychonoff space is Lindelöf if and only if whenever a Tychonoff space $Y$ contains two disjoint closed copies $X_{1}$ and $X_{2}$ of $X$, then these copies can be separated in $Y$ by open sets. We also show that a Tychonoff space $X$ is weakly $C$-embedded (relatively normal) in every larger Tychonoff space if and only if $X$ is either almost compact or Lindelöf (normal almost compact or Lindelöf).

References

A. V. Arhangel′skii, Relative topological properties and relative topological spaces, Proceedings of the International Conference on Convergence Theory (Dijon, 1994), 1996, pp. 87–99. MR 1397067, DOI 10.1016/0166-8641(95)00086-0
A.V.Arhangel’skij and H.M.M.Genedi, Beginnings of the theory of relative topological properties, General Topology, Spaces and Mappings, MGU, Moscow, 1989, pp. 87-89.
A. V. Arhangel′skii and J. Tartir, A characterization of compactness by a relative separation property, Questions Answers Gen. Topology 14 (1996), no. 1, 49–52. MR 1384052
Robert L. Blair and Anthony W. Hager, Extensions of zero-sets and of real-valued functions, Math. Z. 136 (1974), 41–52. MR 385793, DOI 10.1007/BF01189255
Robert L. Blair, On $\upsilon$-embedded sets in topological spaces, TOPO 72—general topology and its applications (Proc. Second Pittsburgh Internat. Conf., Pittsburgh, Pa., 1972; dedicated to the memory of Johannes H. de Groot), Lecture Notes in Math., Vol. 378, Springer, Berlin, 1974., pp. 46–79. MR 0358677
Anthony W. Hager and Donald G. Johnson, A note on certain subalgebras of $C({\mathfrak {X}})$, Canadian J. Math. 20 (1968), 389–393. MR 222647, DOI 10.4153/CJM-1968-035-4
Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
Leonard Gillman and Meyer Jerison, Rings of continuous functions, Graduate Texts in Mathematics, No. 43, Springer-Verlag, New York-Heidelberg, 1976. Reprint of the 1960 edition. MR 0407579
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
K. Kunen, Combinatorics, Handbook of Mathematical Logic (J. Barwise, eds.), Elsevier S.P., North-Holland, Amsterdam, 1977.
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
R. M. Stephenson Jr., Initially $\kappa$-compact and related spaces, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 603–632. MR 776632
S. Watson, The Construction of Topological Spaces: Planks and Resolutions, Recent Progress in General Topology (M. Husek and J. van Mill, eds.), Elsevier S.P., North-Holland, Amsterdam, 1992, pp. 673–757.