Previous |  Up |  Next

Article

Full entry | Fulltext not available (moving wall 24 months)      Feedback
Keywords:
exponential $\kappa$-domination; exponential $\kappa$-cofinality; $\kappa$-stable space; $i$-weight; function space; duality; $\kappa^+$-small diagonal
Summary:
Given a Tychonoff space $X$ and an infinite cardinal $\kappa$, we prove that exponential $\kappa$-domination in $X$ is equivalent to exponential $\kappa$-cofinality of $\,C_p(X)$. On the other hand, exponential $\kappa$-cofinality of $X$ is equivalent to exponential $\kappa$-domination in $C_p(X)$. We show that every exponentially $\kappa$-cofinal space $X$ has a $\kappa^+$-small diagonal; besides, if $X$ is $\kappa$-stable, then $nw(X) \leq \kappa$. In particular, any compact exponentially $\kappa$-cofinal space has weight not exceeding $\kappa$. We also establish that any exponentially $\kappa$-cofinal space $X$ with $l(X) \leq\kappa$ and $t(X) \leq \kappa$ has $i$-weight not exceeding $\kappa$ while for any cardinal $\kappa$, there exists an exponentially $\o$-cofinal space $X$ such that $l(X) \geq \kappa$.
References:
[1] Arkhangel'skiĭ A. V.: Factorization theorems and spaces of functions: stability and monolithism. Dokl. Akad. Nauk SSSR 265 (1982), no. 5, 1039–1043 (Russian). MR 0670475
[2] Arkhangel'skiĭ A. V.: Continuous mappings, factorization theorems and spaces of functions. Trudy Moskov. Mat. Obshch. 47 (1984), 3–21, 246 (Russian). MR 0774944
[3] Arkhangel'skiĭ A. V.: Topological Function Spaces. Mathematics and Its Applications (Soviet Series), 78, Kluwer Academic Publishers Group, Dordrecht, 1992. DOI 10.1007/978-94-011-2598-7_4 | MR 1144519
[4] Asanov M. O.: On cardinal invariants of spaces of continuous functions. Sovr. Topologia i Teoria Mnozhestv 2 (1979), 8–12 (Russian).
[5] Engelking R.: General Topology. Monografie Matematyczne, 60, PWN—Polish Scientific Publishers, Warsaw, 1977. MR 0500780
[6] Gruenhage G., Tkachuk V. V., Wilson R. G.: Domination by small sets versus density. Topology Appl. 282 (2020), 107306, 10 pages. DOI 10.1016/j.topol.2020.107306 | MR 4116835
[7] Hodel R. E.: Cardinal Functions. I. Handbook of Set-Theoretic Topology, North Holland, Amsterdam, 1984, 1–61. MR 0776620
[8] Hušek M.: Topological spaces without $\kappa$-accessible diagonal. Comment. Math. Univ. Carolinae 18 (1977), no. 4, 777–788. MR 0515009
[9] Juhász I., Szentmiklóssy Z.: Convergent free sequences in compact spaces. Proc. Amer. Math. Soc. 116 (1992), no. 4, 1153–1160. DOI 10.2307/2159502 | MR 1137223 | Zbl 0767.54002
[10] Noble N.: The density character of function spaces. Proc. Amer. Math. Soc. 42 (1974), no. 1, 228–233. DOI 10.1090/S0002-9939-1974-0328855-4 | MR 0328855
[11] Pytkeev E. G.: Tightness of spaces of continuous functions. Uspekhi Mat. Nauk 37 (1982), no. 1(223), 157–158 (Russian). MR 0643782
[12] Tkachuk V. V.: A $C_p$-Theory Problem Book. Topological and Function Spaces, Problem Books in Mathematics, Springer, New York, 2011. MR 3024898
[13] Tkachuk V. V.: A $C_p$-Theory Problem Book. Special Features of Function Spaces, Problem Books in Mathematics, Springer, Cham, 2014. MR 3243753
[14] Tkachuk V. V.: A $C_p$-Theory Problem Book. Compactness in Function Spaces, Problem Books in Mathematics, Springer, Cham, 2015. MR 3243753
Partner of
EuDML logo