A space X is called functionally countable if ƒ (X) is countable for any continuous function ƒ : X → Ø. Given an infinite cardinal k, we prove that a compact scattered space K with d(K) > k must have a convergent k+-sequence. This result implies that a Corson compact space K is countable if the space (K × K) \ ΔK is functionally countable; here ΔK = {(x, x): x ϵ K} is the diagonal of K. We also establish that, under Jensen’s Axiom ♦, there exists a compact hereditarily separable non-metrizable compact space X such that (X × X) \ ΔX is functionally countable and show in ZFC that there exists a non-separable σ-compact space X such that (X × X) \ ΔX is functionally countable.
A. V. Arhangel’skii . Topological Function Spaces. Kluwer Acad. Publ., Dordrecht, 1992.
B. Cascales and J. Orihuela . On compactness in locally convex spaces. Math. Z., 195:365–381, 1987.
J. Chaber . Conditions which imply compactness in countably compact spaces. Bull. Acad. Pol. Sci. Ser.Math., 24:993–998, 1976.
G. I. Chertanov . Products of linearly ordered spaces and continuous maps. Dokl. Akad. Nauk SSSR, 223:6:1322–1325, 1975.
A. Dow and D. Guerrero Sánchez . Domination conditions under which a compact space is metrizable. Bull. Austral. Math. Soc., 91:502–507, 2015.
A. Dow and K.P. Hart . Compact spaces with a ℙ-diagonal. Indag. Math., 27(3):721–726, 2016.
F. Galvin . Problem 6444. Amer. Math. Monthly, 90(9):648, 1983. Solution: Amer. Math. Monthly, 92(6):434, 1985.
H. Z. Hdeib and C.M. Pareek . A generalization of scattered spaces. Topology Proc., 14:59–74, 1989.
R. Levy and M. Matveev . Functional separability. Comment. Math. Univ. Carolinae, 51:4:705–711, 2010.
J. T. Moore . A solution to the L space problem. J. Amer. Math. Soc., 19:3:717–736, 2006.
A. Ostaszewski . On countably compact perfectly normal spaces. J. London Math. Soc., 14:505–516, 1976.
A. Pelczynski and Z. Semadeni . Spaces of continuous functions III (Spaces C(Ω) for Ω without perfect subsets). Studia Math., 18:211–222, 1959.
W. Rudin . Continuous functions on compact spaces without perfect subsets. Proc. Amer. Math. Soc., 8:39–42, 1957.
V. V. Tkachuk . A nice subclass of functionally countable spaces. Comment. Math. Univ. Carolinae, 59:3:399–409, 2018.
V. V. Tkachuk . Some applications of exponentially separable spaces. Quaestiones Math., 43:10:1392–1403, 2020.
V. V. Tkachuk . The extent of a weakly exponentially separable space can be arbitrarily large. Houston J. Math., 46:3:809–819, 2020.
V. V. Tkachuk . A Cp -theory Problem Book. Topological and Function Spaces. Springer, New York, 2011.
V. V. Tkachuk . A Cp -theory Problem Book. Special Features of Function Spaces. Springer, New York, 2014.
V. V. Tkachuk . A Cp -Theory Problem Book. Compactness in Function Spaces. Springer, New York, 2015.