A note on condensations of function spaces onto 𝜎-compact and analytic spaces

Krupski, Mikołaj

doi:10.1090/S0002-9939-2015-12507-5

A note on condensations of function spaces onto $\sigma$-compact and analytic spaces
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Proc. Amer. Math. Soc. 143 (2015), 2263-2268 Request permission

Abstract:

Modifying a construction of W. Marciszewski we prove (in ZFC) that there exists a subspace $X$ of the real line $\mathbb {R}$, such that the realcompact space $C_p(X)$ of continuous real-valued functions on $X$ with the pointwise convergence topology does not admit a continuous bijection onto a $\sigma$-compact space. This answers a question of Arhangel’skii.

References

A. V. Arhangel’skii, $C_p$-Theory, in: Recent Progress in General Topology, M. Hušek and J. van Mill (eds.), Elsevier 1992, 1–56.
A. V. Arhangel′skii, On condensations of $C_p$-spaces onto compacta, Proc. Amer. Math. Soc. 128 (2000), no. 6, 1881–1883. MR 1751998, DOI 10.1090/S0002-9939-00-05758-0
A. V. Arkhangel′skiĭ and V. I. Ponomarev, Fundamentals of general topology, Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht, 1984. Problems and exercises; Translated from the Russian by V. K. Jain; With a foreword by P. Alexandroff [P. S. Aleksandrov]. MR 785749
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
D. Lutzer, J. van Mill, and R. Pol, Descriptive complexity of function spaces, Trans. Amer. Math. Soc. 291 (1985), no. 1, 121–128. MR 797049, DOI 10.1090/S0002-9947-1985-0797049-2
Witold Marciszewski, A function space $C_p(X)$ without a condensation onto a $\sigma$-compact space, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1965–1969. MR 1955287, DOI 10.1090/S0002-9939-02-06668-6
Witold Marciszewski, Function spaces, Recent progress in general topology, II, North-Holland, Amsterdam, 2002, pp. 345–369. MR 1970004, DOI 10.1016/B978-044450980-2/50013-3
Henryk Michalewski, Condensations of projective sets onto compacta, Proc. Amer. Math. Soc. 131 (2003), no. 11, 3601–3606. MR 1991774, DOI 10.1090/S0002-9939-03-06882-5
Jan van Mill, The infinite-dimensional topology of function spaces, North-Holland Mathematical Library, vol. 64, North-Holland Publishing Co., Amsterdam, 2001. MR 1851014
V. V. Tkachuk, Condensations of $C_p(X)$ onto $\sigma$-compact spaces, Appl. Gen. Topol. 10 (2009), no. 1, 39–48. MR 2602601, DOI 10.4995/agt.2009.1786
Vladimir V. Tkachuk, A $C_p$-theory problem book, Problem Books in Mathematics, Springer, New York, 2011. Topological and function spaces. MR 3024898, DOI 10.1007/978-1-4419-7442-6