Let X be a compact Hausdorff topological space. We show that multiplication in the algebra C(X) is open iff dim X < 1. On the other hand, the existence of non-empty open sets U,V ⊂ C(X) satisfying Int(U· V) = ∅ is equivalent to dim X > 1. The preimage of every set of the first category in C(X) under the multiplication map is of the first category in C(X) × C(X) iff dim X ≤ 1.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-2-4,
author = {Andrzej Komisarski},
title = {A connection between multiplication in C(X) and the dimension of X},
journal = {Fundamenta Mathematicae},
volume = {189},
year = {2006},
pages = {149-154},
zbl = {1093.54004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-2-4}
}
Andrzej Komisarski. A connection between multiplication in C(X) and the dimension of X. Fundamenta Mathematicae, Tome 189 (2006) pp. 149-154. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-2-4/