Some Questions and References on Relative Topological Properties, Part 1

A.V. Arhangel'skii

Questions

Problem 2. Let Y be a Hausdorff space that is internally regular in every larger Hausdorff space X. Is then Y compact (in itself)?

Problem 3. Let Y be a subspace of a regular space X such that Y is compact in X from inside. Is then Y Tychonoff?

Problem 4. Give an example of a space X and of its subspace Y such that Y is normal in X from inside, but Y is not internally normal in X.

Problem 5. Let Y be a subspace of a Urysohn space X such that Y is compact in X from inside. Is then Y Tychonoff?

Problem 6. Let Y be a subspace of a regular space X such that Y is normal in X. Is then Y Tychonoff?

Problem 7. Let Y be a subspace of a regular space X such that Y is internally normal in X. Is then Y Tychonoff?

Problem 8. Let Y be a subspace of a regular space X such that Y is metrizable in X and Y is Lindelöf in X. Is then Y separable (in itself, in X)?

Problem 9. Is it consistent with ZFC that every densely normal locally compact Hausdorff space X is densely collectionwise normal?

Problem 10. Is it consistent with ZFC that every densely normal first countable regular space is densely collectionwise normal?

Problem 11. Is it true that every densely collectionwise normal regular space with a uniform base is metrizable?

Problem 12. Let Y be a subspace of a Cech-complete space X such that Y is paracompact (1-paracompact) in X. Does there exists a subspace Z of X such that Y is a subset of Z and Z is paracompact and Cech-complete?

Problem 13. Let Y be a (dense) subspace of a Tychonoff space X such that X is normal and countably paracompact (1-countably paracompact) on Y. Is then true that X x I is normal on Y x I? (Where I is the closed interval [0, 1]). Is Y x I internally normal in X x I?

Problem 14. Let Y be a (dense) subspace of a Tychonoff space X such that X is normal on Y and let Z be a countable Tychonoff space. Is then true that X x Z is normal on Y x Z?

Problem 15. Let Y be a subspace of a regular space X such that Y is 2^\omega-compact in X and X has countable tightness. Is then the closure of Y in X compact? What if, in addition, X is Tychonoff and first countable?

Problem 16. Let Y be a (dense) subspace of a Tychonoff space X such that X x B is normal on Y x B, for each compact Hausdorff space B. Is then Y paracompact (1-paracompact) in X?

Problem 17. Let Y be a (dense) subspace of a Tychonoff space X such that Y is internally normal in X and bounded in X. Is then Y countably compact in X?

Problem 18. Let Y be a (dense) subspace of a Tychonoff space X such that Y x I is internally normal in X x I. Is then Y countably paracompact 1-countably paracompact) in X?

Problem 19. Let Y be a (dense) pseudocompact subspace of a Tychonoff space X such that Y is internally normal in X. Is then Y countably compact in X?

References

[1]

Arhangel'skii A.V., Relative topological properties and relative topological spaces. Topology and Appl. 70 (1996), 1-13.

[2]

Arhangel'skii A.V., Properties of placement type: relative strong pseudocompactness. Proc. Steklov Inst. Math. 3 (1993), 25-27.

[3]

Arhangel'skii A.V. and H.M.M. Genedi, Location of subspaces in spaces; relative versions of compactness, lindelöf property and separation axioms. Moscow Univ. Math. Bull. 44 (1989), 67-69.

[4]

Arhangel'skii A.V. and I.Ju. Gordienko, Relatively locally finite Hausdorff spaces. Questions and Answers in General Topology 12 (1994), 15-25.

[5]

Arhangel'skii A.V. and I.Ju. Gordienko, On relative normality and relative symmetrizability. Moscow Univ. Math. Bull. 50:3 (1995).

[6]

Arhangel'skii A.V. and I.Ju. Gordienko, Relative symmetrizability and metrizability. Comment. Math. Univ. Carol. 37:4 (1996), 757-774.

[7]

Arhangel'skii A.V. and I.V. Yaschenko, Relatively compact spaces and separation properties. Comment. Math. Univ. Carol. 37:2 (1996), 343-348.

[8]

Arhangel'skii A.V. and T. Nogura, Relative sequentiality. Topology and Appl. 82 (1998), 49-58.

[9]

Arhangel'skii A.V. and J. Tartir, A characterization of compactness by a relative separation property. Questions Answers general Topology 14:1 (1996).

[10]

Dow A., and J. Vermeer, An example concerning the property of a space being Lindelöf in another. Topology and Appl. 51 (1993), 255-260.

[11]

Gordienko I.Ju., On relative topological properties of normality type. Mosc. Univ. Math. Bull. 5 (1992).

[12]

Hart K.P., and J. Vermeer, Non-Tychonoff e-compactifiable spaces. Proc. AMS 89:4 (1983), 725-729.

[13]

Hechler S.H., On a notion of weak compactness in non-regular spaces. Studies in Topology, Allen K.R., Stavrakas N.M., Ed-rs. Academic Press, New York (1975), 215-237.

[14]

Just W. and J. Tartir, A \kappa-normal, not densely normal Tychonoff space. Proc. AMS (199?).

[15]

Matveev M.V., Pavlov O.I., and J.K. Tartir, On relatively normal spaces, relatively regular spaces, and on relative property (a). To appear in Topology and Appl. (199?).

[16]

Rancin, Compactness modulo an ideal. Soviet Math. Dokl. 13:1 (1972), 193-197.

Ohio University
Athens, OH, U.S.A.
arhangel@bing.math.ohiou.edu
http://www.math.ohiou.edu/~arhangel

Moscow State University
Moscow, Russia
arhala@arhala.mccme.ru
http://mech.math.msu.su/department/gentopol/arhangel.html

Date received: June 15, 2000.