Some Realcompact Spaces
Keywords:
$C^*$-embedding, $C$-embedding, copy of $\mathbb{N}$, realcompactAbstract
We present examples of realcompact spaces with closed subsets that are $C^*$-embedded but not $C$-embedded, including one where the closed set is a copy of $\mathbb{N}$.
References
Doyel Barman, Alan Dow, and Roberto Pichardo-Mendoza, Complete separation in the random and Cohen models, Topology Appl. 158 (2011), no. 14, 1795-1801.
Jean Dieudonné, Une généralisation des espaces compacts, J. Math. Pures Appl. (9) 23 (1944), 65-76.
Alan Dow, Klaas Pieter Hart, Jan van Mill, and Hans Vermeer, Closed copies of $\mathbb{N}$ in $\mathbb{R}^{\omega_1}$. Available at https://arxiv.org/abs/ 2307.07223 [math.GN].
Ryszard Engelking, General Topology. Translated from the Polish by the author. 2nd ed. Sigma Series in Pure Mathematics, 6. Berlin: Heldermann Verlag, 1989.
Leonard Gillman and Meyer Jerison, Rings of Continuous Functions. Graduate Texts in Mathematics, No. 43. New York-Heidelberg: Springer-Verlag, 1976.
M. Katětov, On real-valued functions in topological spaces, Fund. Math. 38 (1951), 85-91.
Haruto Ohta and Kaori Yamazaki, Extension problems of real-valued continuous functions in Open Problems in Topology II. Ed. Elliott Pearl. Amsterdam: Elsevier, 2007. 35-45.
Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. New York-Heidelberg: Springer-Verlag, 1978.
A. Tychonoff, Über die topologische Erweiterung von Räumen, Math. Ann. 102 (1930), no.1, 544-561.
