Problems I and my students couldn't solve
Keywords:
Generalized Souslin Hypothesis, perfectly normal non-archimedian, collectionwise normal, $k$-space, Lindelöf, elementary submodel topology, productively Lindelöf, moving off propertyAbstract
Here is a selection of problems in set-theoretic topology which I think are interesting, important, and which I and/or my students tried and failed to solve.
References
K . Alster. On the class of all spaces of weight not greater than $\omega_1$ whose Cartesian product with every Lindelöf space is Lindelöf. Fund. Math., 129:133-140, 1988.
A. Arhangel'skii. On the cardinality of bicompacta satisfying the first axiom of countability. Soviet Math. Dokl., 10:951-955, 1969.
A. Arhangel'skii. Topological function spaces. Kluwer Academic Publishers, 1992.
L. F. Aurichi and L. R. Zdomskyy. Internal characterizations of productively Lindelöf spaces. Proc. Amer. Math. Soc., 146(8):3615-3626, 2018.
Z. Balogh. On the collectionwise normality of locally compact normal spaces. Trans. Amer. Math. Soc., 323:389-411, 1991.
J. E. Baumgartner. Surveys in Set Theory, volume 87 of London Mathematical Society Lecture Note Series, chapter Iterated Forcing, pages 1-59. Cambridge University Press, 1983.
Z. Balogh, S. Davis, A. Dow, G. Gruenhage, P. Nyikos, M. E. Rudin, F. D. Tall, and S. Watson. New Classic Problems. Topology Proc., 15:201-220, 1990.
H. Bennett, R. Heath, and D. Lutzer. Generalized ordered spaces with $\sigma$-closed-discrete dense subsets. Proc. Amer. Math. Soc., 129:931-939, 2000.
R. H. Bing. Metrization of topological spaces. Canad. J. Math, 3:175-186, 1951.
H. Bennett and D. Lutzer. Problems in perfect ordered spaces. In G. M. Reed and J. van Mill, editors, Open Problems in Topology, pages 232-236. North-Holland, Amsterdam, 1990.
J. E. Baumgartner and F. D. Tall. Reflecting Lindelöfness. Topology Appl., 122:35-49, 2002.
A. Charlesworth, R. Hodel, and F. Tall. On a theorem of Jones and Heath concerning separable normal spaces. Colloq. Math., 34:33-37, 1975.
P. Daniels. Normal $k^{\prime}$-spaces are consistently collectionwise normal. Fund. Math., 138:225-234, 1991.
K. J. Devlin. The combinatorial principle $\diamond^{\#}$. J. Symbolic Logic, 47:888-899, 1982.
A. Dow, I. Juhász, and W. Weiss. Integer-valued functions and increasing unions of first countable spaces. Israel J. Math., 67:181-192, 1989.
A. Dow. Two applications of reflection and forcing to topology. In Z. Frolík, editor, General Topology and its Relation to Modern Analysis and Algebra VI, Sixth Prague Topological Symposium 1986, pages 155-172, Heldermann Verlag, Berlin, 1988.
A. Dow. Set Theory in Topology. In M. Hušek and J. van Mill, editors, Recent Progress in General Topology, pages 167-197. North Holland, 1992.
A. Dow. A new Lindelöf space with points $G_\delta$. Comment. Math. Univ. Carolin., 56(2):223-230, 2015.
A. Dow and F. D. Tall. PFA $(S)[S]$ and countably compact spaces. Topology Appl., 230:393-416, 2017.
A. Dow, F. Tall, and W. Weiss. New proofs of the consistency of the normal Moore space conjecture, I. Topology Appl., 37:35-51, 1990.
A. Dow, F. Tall, and W. Weiss. New proofs of the consistency of the normal Moore space conjecture, II. Topology Appl., 37:115-129, 1990.
W. G. Fleissner. Handbook of Set-theoretic Topology, chapter The normal Moore space conjecture and large cardinals, pages 733-760. North-Holland, Amsterdam, 1984.
W. G. Fleissner. Theorems from Measure Axioms, Counterexamples from $\diamond^{++}$. In F. D. Tall, editor, The work of Mary Ellen Rudin, volume 705 of Annals of the New York Academy of Sciences, pages 66-77. 1993.
G. Fernandes, G. Pinto, and V. Rocha. On totally Lindelöf spaces. Topology Appl., 341, 2024. Article 108704.
R. Grunberg, L. R. Junqueira, and F. D. Tall. Forcing and normality. Topology Appl., 84:145-174, 1998.
G. Gruenhage and D. K. Ma. Baireness of $C_k(X)$ for locally compact $X$. Topology Appl., 80(1):131-139, 1997.
I. Gorelic. The Baire category and forcing large Lindelöf spaces with points $G_\delta$. Proc. Amer. Math. Soc., 118(2):603-607, 1993.
G. Gruenhage. The story of a topological game. Rocky Mt. J. Math., 36:18851914, 2006.
R. W. Heath. Screenability, pointwise paracompactness, and metrization of moore spaces. Canad. J. Math., 16:763-770, 1964.
R. W. Heath. Separability and $\aleph_1$-compactness. Colloq. Math., 12:11-14, 1964.
A. Hajnal and I. Juhász. Remarks on the cardinality of compact spaces and their Lindelöf subspaces. Proc. Amer. Math. Soc., 59:146-148, 1976.
F. B. Jones. Concerning normal and completely normal spaces. Bull. Amer. Math. Soc., 43:671-677, 1937.
L. R. Junqueira, P. Larson, and F. D. Tall. Compact spaces, elementary submodels, and the countable chain condition. Ann. Pure Appl. Logic, 144:107-116, 2006.
L. R. Junqueira and F. D. Tall. The topology of elementary submodels. Topology Appl., 82:239-266, 1998.
I. Juhász and W. A. R. Weiss. On a problem of Sikorski. Fund. Math., 100:223227, 1978.
P. Koszmider and F. D. Tall. A Lindelöf space with no Lindelöf subspaces of size $\aleph_1$. Proc. Amer. Math. Soc., 130:2777-2787, 2002.
H. Lamei Ramandi and S. Todorcevic. Can you take Komjath's inaccessible away? Ann. Pure Appl. Logic, 175(7), 2024. Article 103452.
D. K. Ma. The Cantor tree, the $\gamma$-property, and Baire function spaces. Proc. Amer. Math. Soc., 119:903-913, 1993.
W. Mitchell. Aronszajn trees and the independence of the transfer property. Ann. Math. Logic, 5:21-46, 1972.
R. A. McCoy and I. Ntantu. Topological properties of spaces of continuous functions. Number 1315 in Lecture Notes in Mathematics. Springer, 1988.
P. J. Nyikos. A Survey of Zero-Dimensional Spaces. In S. Franklin and B. S. Thomas, editors, Proc. Ninth Annual Spring Topology Conf., pages 87-114, Memphis State Univ., Memphis, Tenn., 1975.
P. J. Nyikos. A provisional solution to the normal Moore space problem. Proc. Amer. Math. Soc., 78(3): 429-435,1980.
A. Ostaszewski. On countably compact, perfectly normal spaces. J. Lond. Math. Soc., 14(2):505-516, 1976.
R. G. A. Prado. Applications of reflection to topology. PhD thesis, University of Toronto, 1999.
Y.-Q. Qiao. Martin's Axiom does not imply perfectly normal non-Archimedean spaces are metrizable. Proc. Amer. Math. Soc., 129:1179-1188, 2000.
Y.-Q. Qiao and F. D. Tall. Perfectly normal non-Archimedean spaces in Mitchell models. Topology Proc., 18:231-244, 1993.
Y.-Q. Qiao and F. D. Tall. Perfectly normal non-metrizable non-Archimedean spaces are generalized Souslin lines. Proc. Amer. Math. Soc., 131(12):3929-3936, 2003.
M. Scheepers. Measurable cardinals and the cardinality of Lindelöf spaces. Topology Appl., 157:1651-1657, 2010.
L. J. Stanley and S. Shelah. S-forcing. I. A "black-box" theorem for morasses, with applications to super-Souslin trees. Israel J. Math., 43(3):185-224, 1982.
S. Shelah and S. B. Todorcevic. A note on small Baire spaces. Canad. J. Math., 38(3):659-665, 1986.
M. Scheepers and F. D. Tall. Lindelöf indestructibility, topological games and selection principles. Fund. Math., 210:1-46, 2010.
F. D. Tall. Set-theoretic consistency results and topological theorems concerning the normal Moore space conjecture and related results. PhD thesis, University of Wisconsin (Madison), 1969.
F. D. Tall. On the cardinality of Lindelöf spaces with points $G_\delta$. Topology Appl., 63:21-38, 1995.
F. D. Tall. If it looks and smells like the reals .... Fund. Math., 163:1-11, 2000.
F. D. Tall. An irrational problem. Fund. Math., 175:259-269, 2002.
_ Set-theoretic problems concerning Lindelöf spaces.
Quest. Ans. Gen. Top., 29:91-103, 2011. Available online at https://www.math.toronto.edu/tall/publications/Lindelof.pdf.
F. D. Tall. Some observations on the Baireness of $C_k(X)$ for a locally compact space X. Topology Appl., 213:212-219, 2016.
V. V. Tkachuk. A $C_p$-Theory Problem Book I-IV. Problem Books in Mathematics. Springer, 2011-2015.
S. Todorčević. Some Consequences of MA+ ⇁wKH. Topology Appl., 12:187202, 1981.
S. Todorčević. Trees, subtrees, and order types. Ann. Math. Logic, 20:233-268, 1981.
F. D. Tall and T. Usuba. Lindelöf spaces with small pseudocharacter and an analog of Borel's conjecture for subsets of $[0,1]^{\aleph_1}$. Houston J. Math., 40:12991309, 2014.
T. Usuba. Large regular Lindelöf spaces with points $G_\delta$. Fund. Math., 237:249260, 2017.
D. J. Velleman. Morasses, diamond, and forcing. Ann. Math. Logic, 23(2-3):199281, 1982.
P. D. Welch. On possible non-homeomorphic substructures of the real line. Proc. Amer. Math. Soc., 130(9) :2771-2775, 2002.
