Lindelöf property and absolute embeddings

Authors:
A. Bella and I. V. Yaschenko

Journal:
Proc. Amer. Math. Soc. **127** (1999), 907-913

MSC (1991):
Primary 54A35, 54D20

MathSciNet review:
1469399

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

**[Ar]**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**, 10.1016/0166-8641(95)00086-0**[AG]**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.**[AT]**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****[BH]**Robert L. Blair and Anthony W. Hager,*Extensions of zero-sets and of real-valued functions*, Math. Z.**136**(1974), 41–52. MR**0385793****[Bl]**Robert L. Blair,*On 𝜐-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), Springer, Berlin, 1974., pp. 46–79. Lecture Notes in Math., Vol. 378. MR**0358677****[HJ]**Anthony W. Hager and Donald G. Johnson,*A note on certain subalgebras of 𝐶(𝔛)*, Canad. J. Math.**20**(1968), 389–393. MR**0222647****[En]**Ryszard Engelking,*General topology*, PWN—Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. MR**0500780****[GJ]**Leonard Gillman and Meyer Jerison,*Rings of continuous functions*, Springer-Verlag, New York-Heidelberg, 1976. Reprint of the 1960 edition; Graduate Texts in Mathematics, No. 43. MR**0407579****[He]**Edwin Hewitt,*A note on extensions of continuous real functions*, Anais Acad. Brasil. Ci.**21**(1949), 175–179. MR**0031711****[Ku]**K. Kunen,*Combinatorics*, Handbook of Mathematical Logic (J. Barwise, eds.), Elsevier S.P., North-Holland, Amsterdam, 1977.**[Sm]**Yu. M. Smirnov,*Mappings of systems of open sets*, Mat. Sbornik N.S.**31(73)**(1952), 152–166 (Russian). MR**0050263****[St]**R. M. Stephenson Jr.,*Initially 𝜅-compact and related spaces*, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 603–632. MR**776632****[Wa]**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. CMP**93:15**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
54A35,
54D20

Retrieve articles in all journals with MSC (1991): 54A35, 54D20

Additional Information

**A. Bella**

Affiliation:
Dipartimento di Matematica, Citta Universitaria, Viale A.Doria 6, 95125, Catania, Italy

Email:
bella@dipmat.unict.it

**I. V. Yaschenko**

Affiliation:
Moscow Center for Continuous Mathematical Education, B.Vlas’evskij per. 11, 121002, Moscow, Russia

Email:
ivan@mccme.ru

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-04568-2

Keywords:
Lindel\"{o}f space,
normal space,
relative topological property,
embedding,
almost compact space

Received by editor(s):
November 14, 1996

Received by editor(s) in revised form:
June 26, 1997

Additional Notes:
This work was done while the second author was visiting Catania University. He is grateful to Italian colleagues for generous hospitality and to CNR for financial support.

Communicated by:
Alan Dow

Article copyright:
© Copyright 1999
American Mathematical Society