We present a forcing construction of a Hausdorff zero-dimensional Lindelöf space $X$ whose square $X^2$ is again Lindelöf but its cube $X^3$ has a closed discrete subspace of size ${\frak c}^+$, hence the Lindelöf degree $L(X^3) = {\frak c}^+ $. In our model the Continuum Hypothesis holds true. After that we give a description of a forcing notion to get a space $X$ such that $L(X^n) = \aleph_0$ for all positive integers $n$, but $L(X^{\aleph_0}) = {\frak c}^+ = \aleph_2$.
@article{118678,
author = {Isaac Gorelic},
title = {On powers of Lindel\"of spaces},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {35},
year = {1994},
pages = {383-401},
zbl = {0815.54015},
mrnumber = {1286586},
language = {en},
url = {http://dml.mathdoc.fr/item/118678}
}
Gorelic, Isaac. On powers of Lindelöf spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) pp. 383-401. http://gdmltest.u-ga.fr/item/118678/
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