Geodesic
A connection between multiplication in $C(X)$ and the dimension of $X$
Andrzej Komisarski1
1 Department of Probability Theory and Statistics Faculty of Mathematics University of Łódź Banacha 22 90-238 Łódź, Poland
Fundamenta Mathematicae, Tome 189 (2006) no. 2, pp. 149-154.

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

Let $X$ be a compact Hausdorff topological space. We show that multiplication in the algebra $C(X)$ is open iff $\dim X1$. On the other hand, the existence of non-empty open sets $U,V\subset C(X)$ satisfying ${\rm Int}(U\cdot V)=\emptyset$ 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 $C(X)\times C(X)$ iff $\dim X \leq 1$.
DOI : 10.4064/fm189-2-4
Keywords: nbsp compact hausdorff topological space multiplication algebra dim other existence non empty sets subset satisfying int cdot emptyset equivalent dim preimage every set first category under multiplication map first category times dim leq

Andrzej Komisarski 1

1 Department of Probability Theory and Statistics Faculty of Mathematics University of Łódź Banacha 22 90-238 Łódź, Poland 
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Andrzej Komisarski. A connection between  multiplication in $C(X)$ 
and the dimension of $X$. Fundamenta Mathematicae, Tome 189 (2006) no. 2, pp. 149-154. doi : 10.4064/fm189-2-4. http://geodesic.mathdoc.fr/articles/10.4064/fm189-2-4/

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