We prove that there exists an example of a metrizable non-discrete space $X$, such that $C_p(X\times \omega )\approx_{l} C_p(X)$ but $C_p(X\times S) \not\approx_{l} C_p(X)$ where $S = (\{0\}\cup\{\frac{1}{n+1}:n\in\omega \})$ and $C_p(X)$ is the space of all continuous functions from $X$ into reals equipped with the topology of pointwise convergence. It answers a question of Arhangel'skii ([2, Problem 4]).
@article{119269,
author = {Henryk Michalewski},
title = {An answer to a question of Arhangel'skii},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {42},
year = {2001},
pages = {545-550},
zbl = {1053.54025},
mrnumber = {1860243},
language = {en},
url = {http://dml.mathdoc.fr/item/119269}
}
Michalewski, Henryk. An answer to a question of Arhangel'skii. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 545-550. http://gdmltest.u-ga.fr/item/119269/
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