A connection between multiplication in $C(X)$ and the dimension of $X$
Volume 189 / 2006
Fundamenta Mathematicae 189 (2006), 149-154
MSC: Primary 54C35; Secondary 54F45, 46E25.
DOI: 10.4064/fm189-2-4
Abstract
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 $U,V\subset C(X)$ 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 $\dim X \leq 1$.
