A note on preimages of countable-to-one continuous maps
Keywords:
countable-to-one continuous map, $Q$-set, $\Delta$-set, Eberlein compact, effectively $\Delta$-spaceAbstract
Let $\mathcal{C}$ be a class of topological spaces and suppose that $f: X \rightarrow Y$ is a continuous countable-to-one surjective map, does $Y \in \mathcal{C}$ imply that $X \in \mathcal{C}$ ?
We consider this question for various classes $\mathcal{C}$ including the class of $\Delta_1$-spaces and its subclasses. One of the main results obtained in this paper says that if $X$ is a compact space and $Y$ is a scattered Eberlein compact space then so is $X$.
We examine various consistent examples of continuous 2-to-1 surjective maps $f: X \rightarrow Y$ such that $Y$ is a $Q$-set of reals, but $X$ is not a $Q$-space. Several open questions which we pose in the paper are surprisingly related to the old and still unsettled problem asking whether there is a $\Delta$-set of reals which is not a $Q$-set.
References
A. Avilés, A. Poveda, S. Todorcevic, Rosenthal compacta that are premetric of finite degree, Fund. Math. 239 (2017), 259-278.
Z. Balogh, There is a paracompact Q-set space in ZFC, Proc. Amer. Math. Soc. 126 (1998), 1827-1833.
E. K. van Douwen, Orderability of all noncompact images, Topology Appl. 51 (1993), 159-172.
R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
J. C. Ferrando, J. Kąkol, A. Leiderman, S. A. Saxon, Distinguished $C_p(X)$ spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115:27 (2021).
G. Gruenhage, $A$ survey of $D$-spaces, Set theory and its applications, Contemporary Math. 533 (2011), 13-28.
M. Hrušák, Á. Tamariz-Mascarúa, M. Tkachenko (Editors), Pseudocompact Topological Spaces, A Survey of Classic and New Results with Open Problems, Springer, 2018.
J. Kąkol and A. Leiderman, A characterization of $X$ for which spaces $C_p(X)$ are distinguished and its applications, Proc. Amer. Math. Soc., series B, 8 (2021), 86-99.
J. Kąkol and A. Leiderman, Basic properties of $X$ for which the space $C_p(X)$ is distinguished, Proc. Amer. Math. Soc., series B, 8 (2021), 267-280.
J. Kąkol, O. Kurka and A. Leiderman, On Asplund spaces $C_k(X)$ and $w^*$ binormality, Results in Mathematics, 78:203 (2023).
J. Kąkol, O. Kurka and A. Leiderman, Some classes of topological spaces extending the class of $\Delta$-spaces, Proc.
Amer. Math. Soc. 152 (2024), 883-899.
J. Kąkol, A. Leiderman and V. V. Tkachuk, Some applications of the $\Delta_1$-property, 2024 (to appear in Topology Appl.)
R. W. Knight, $\Delta$-Sets, Trans. Amer. Math. Soc. 339 (1993), 45-60.
W. Kubiś, A. Moltó, Finitely fibered Rosenthal compacta and trees, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 105 (2011), 23-37.
A. Leiderman and P. Szeptycki, On $\Delta$-spaces, Israel Journal of Mathematics, TBD (2025), 1-30, DOI: 10.1007/s11856-025-2733-2
https://arxiv.org/pdf/2307.16047.pdf
A. Leiderman and V. V. Tkachuk, Pseudocompact $\Delta$-spaces are often scattered, Monatsh. Math. 197 (2022), 493-503.
P. Memarpanahi and P. Szeptycki, $Q$-sets, $\Delta$-sets and $L$-spaces, preprint, https://arxiv.org/abs/2501.08298v2
H. P. Rosenthal, The heredity problem for weakly compactly generated Banach spaces, Compos. Math. 28 (1974), 88-111.
V. V. Tkachuk, A note on $\kappa$-Fréchet-Urysohn property in function spaces, Bull. Belg. Math. Soc. Simon Stevin 28 (2021), 123-132.
S. Todorcevic, Compact subsets of the first Baire class, J. Amer. Math. Soc. 12 (1999), 1179-1212.
