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# Article

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Keywords:
functionally countable; pseudo-$\aleph_1$-compact; DCCC; P-space; $\tau$-simple; scattered; 1-functionally separable; 2-functionally separable; 3-functionally separable; pseudocompact; dyadic compactum; $\sigma$-centered base; LOTS
Summary:
A space $X$ is functionally countable (FC) if for every continuous $f:X\to \mathbb R$, $|f(X)|\leq \omega$. The class of FC spaces includes ordinals, some trees, compact scattered spaces, Lindelöf P-spaces, $\sigma$-products in $2^\kappa$, and some L-spaces. We consider the following three versions of functional separability: $X$ is 1-FS if it has a dense FC subspace; $X$ is 2-FS if there is a dense subspace $Y\subset X$ such that for every continuous $f:X\to \mathbb R$, $|f(Y)|\leq\omega$; $X$ is 3-FS if for every continuous $f:X\to \mathbb R$, there is a dense subspace $Y\subset X$ such that $|f(Y)|\leq \omega$. We give examples distinguishing 1-FS, 2-FS, and 3-FS and discuss some properties of functionally separable spaces.
References:
[1] Arhangel'skii A.V.: Topological properties of function spaces: duality theorems. Soviet Math. Doc. 269 (1982), 1289–1292. MR 0705371
[2] Arhangel'skii A.V.: Topological Function Spaces. Kluwer Academic Publishers, 1992. MR 1485266
[3] Barr M., Kennison F., Raphael R.: Searching for absolute $\mathcal CR$-epic spaces. Canad. J. Math. 59 (2007), 465–487. DOI 10.4153/CJM-2007-020-9 | MR 2319155
[4] Barr M., Burgess W.D., Raphael R.: Ring epimorphisms and $C(X)$. Theory Appl. Categ. 11 (2003), no. 12, 283–308. MR 1988400 | Zbl 1042.54007
[5] Burgess W.D., Raphael R.: Compactifications, $C(X)$ and ring epimorphisms. Theory Appl. Categ. 16 (2006), no. 21, 558–584. MR 2259263 | Zbl 1115.18001
[6] van Douwen E.K.: Density of compactifications. Set-theoretic Topology, Academic Press, New York, 1977, pp. 97-110. MR 0442887 | Zbl 0379.54006
[7] Galvin F.: Problem 6444. Amer. Math. Monthly 90 (1983), no. 9, 648; solution: Amer. Math. Monthly 92 (1985), no. 6, 434.
[8] Hrušák M., Raphael R., Woods R.G.: On a class of pseudocompact spaces derived from ring epimorphisms. Topology Appl. 153 (2005), 541–556. MR 2193326
[9] Levy R., Rice M.D.: Normal $P$ spaces and the $G_\delta$-topology. Colloq. Math. 44 (1981), 227–240. MR 0652582 | Zbl 0496.54034
[10] Matveev M.: One more topological equivalent of CH. Topology Appl. 157 (2010), 1211–1214. DOI 10.1016/j.topol.2010.02.013 | MR 2607088 | Zbl 1190.54003
[11] Moore J.T.: A solution to the $L$ space problem. J. Amer. Math. Soc. 19 (2006), no. 3, 717–736. DOI 10.1090/S0894-0347-05-00517-5 | MR 2220104 | Zbl 1107.03056
[12] Moore J.T.: An $L$ space with a $d$-separable square. Topology Appl. 155 (2008), 304–307. DOI 10.1016/j.topol.2007.07.006 | MR 2380267 | Zbl 1146.54015
[13] Noble N., Ulmer M.: Factorizing functions on cartesian products. Trans. Amer. Math. Soc. 163 (1972), 329–339. DOI 10.1090/S0002-9947-1972-0288721-2 | MR 0288721
[14] Pełczyński A., Semadeni Z.: Spaces of continuous functions III. Spaces $C(\Omega)$ for $\Omega$ without perfect subsets. Studia Math. 18 (1959), 211–222. MR 0107806
[15] Raphael M., Woods R.G.: The epimorphic hull of $C(X)$. Topology Appl. 105 (2002), 65–88. MR 1761087 | Zbl 1069.18001
[16] Reznichenko E.A.: A pseudocompact space in which only sets of complete cardinality are not closed and not discrete. Moscow Univ. Math. Bull. (1989), no. 6, 69–70. MR 1065983
[17] Rudin W.: Continuous functions on compact spaces without perfect subsets. Proc. Amer. Math. Soc. 8 (1957), 39–42. DOI 10.1090/S0002-9939-1957-0085475-7 | MR 0085475 | Zbl 0077.31103
[18] Steprans J.: Trees and continuous mappings into the real line. Topology Appl. 12 (1981), no. 2, 181–185. DOI 10.1016/0166-8641(81)90019-5 | MR 0612014 | Zbl 0457.54010

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