Let $X$ and $Y$ be two topological spaces with $C(X) \cong C(Y)$ (where $C(X)$ is the ring of all continuous real valued functions on $X$). I know that we can not conclude that $X$ and $Y$ are homeomorphic. But I wonder how independent $X$ and $Y$ could be ? For example is there any forced relation between their cardinality ?



Isomorphic rings of functions
Let $X,Y$ be arbitrary sets (of arbitrary cardinals) armed with topologies $\tau\_1 = \lbrace \emptyset, X\rbrace$ and $\tau\_2 = \lbrace \emptyset, Y\rbrace$. Then it is clear that $C(X) \cong \Bbb{R} \cong C(Y)$. So there is no forced relation between the cardinal numbers. 


