Some References and Open Problems on Condensations

A.V. Arhangel'skii

Open Problems

Problem 2. Is it possible to condense every complete metric space onto a \sigma-compact Hausdorff space?

Problem 3. Is it possible to condense every topological group metrizable by a complete metric onto a compact Hausdorff space? Onto a \sigma-compact space?

Problem 4. Let X be a separable metrizable space. Is it possible to condense C_p(X) onto a compact space? Onto a \sigma-compact space? What if X is the space J of irrational numbers?

Problem 5. Let X be a compact Hausdorff space. C_p(X) onto a \sigma-compact space?

Problem 6. Is there in ZFC a Lindelöf (a normal) Tychonoff space X of cardinality \omega₁ such that its square X x X can not be condensed onto a Lindelöf (onto a normal) space?

Problem 7. Is it possible to condense every separable metrizable space onto a \sigma-compact space?

Problem 8. Let X be a compact Hausdorff space without isolated points. Is it possible to find two disjoint dense subspaces Y and Z of X such that Y condenses onto a compact Hausdorff space and Z condenses onto a compact Hausdorff space? Can we always do it in such a way that X is the union of Y and Z?

Problem 9. Let X be a Tychonoff space. Can one find a dense subspace Y of X such that Y condenses onto a normal space? Onto a \kappa-normal space? Onto a space Z which contains a dense normal subspace?

Problem 10. Is it true that every Cech-complete space X contains a dense subspace Y such that Y condenses onto a compact space? What if X is a Cech-complete Lindelöf space?

Problem 11. Is it true that every Cech-complete Lindelöf space X condenses onto a \sigma-compact (Tychonoff, Hausdorff) space?

References

[1]

Arhangel'skii A.V., On spread and condensations. Proc. AMS 124 (1996), 3519-3527.

[2]

Banach S., Livre Ecossais. Problem 1, 17 8. 1935; See also Colloq. Math. 1:2 (1948), 150.

[3]

Buzyakova R.Z., On the product of normal spaces. Moscow Univ. Math. Bull. 49:5 (1994), 52-53.

[4]

Buzyakova R.Z., On condensations of Cartesian products onto normal spaces. Moscow Univ. Math. Bull. 51:1 (1996), 13-14.

[5]

Buzyakova R.Z., An example of two normal groups the product of which does not condense onto a normal space. Moscow Univ. Math. Bull. 52:3 (1997), 42

[6]

Buzyakova R.Z., A criterion that a pseudocompact space condenses onto a compact space. Questions and Answers in General Topology (199?).

[7]

Chen Y.Q., Note on two questions of Arhangelskii. Accepted by Questions and Answers in General Topology (1998).

[8]

van Douwen E.K., A technique for constructing honest locally compact submetrizable examples. Topology Appl. 47 (1992), 179-201.

[9]

Kulpa W., On a Problem of Banach. Colloq. Math. 56:2 (1988), 255-261.

[10]

Nadler S.B., Quinn J., and H. Reiter, Results and problems concerning compactifications, compact subtopologies, and mappings. Fund. Math. 89 (1975), 33-44.

[11]

Pavlov O., Condensations of finite products. A preprint.

[12]

Pavlov O., Condensations of Cartesian products. A preprint.

[13]

Pavlov O., A small Tychonoff space which can not be mapped one-to-one onto any normal space. A preprint.

[14]

Pytkeev E.G., The upper bounds of topologies. Mat. Zametki 20 (1976), 489-500. Translation: Math. Notes 20 (1976), 831-837.

[15]

Pytkeev E.G., On the theory of condensations onto compacta. Soviet Mat. Dokl. 18 (1977).

[16]

Yascenko I.V., Cardinality of discrete families of open sets and one-to-one continuous mappings. Questions and Answers in General Topology 10 (1992), 75-78.

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.