We denote The ringsthe ring of all continuous Real valuedreal-valued functions on $X$ by $C(X)$. also The ring of all bounded continuous real valued functions on $X$ is denoted by $C_b(X)$. One of the goals of the study of $C(X)$ is to connect algebraic properties of the Ringring $C(X)$ with the topological properties of the space $X$.
For a simple and well-Knownknown connection we could seethere is the following theorem:
Theorem: theThe topological space $X$ is connected iff the ring $C(X)$ has no idempotent element other than $0$ and $1$.
My Questionquestion comes from the relation between ring homomorphisms of the rings $C(X)$, $C(Y)$ and topological spaces $X,Y$.
$Q_1$:Let $X$ and, $Y$ be two $T_{3\frac{1}{2}}$topological spaces. If
Q1: Let $X$ and $Y$ be two $T_{3\frac{1}{2}}$topological spaces. If $X$ is a zero dimensional space and $C(X)$ is ring isomorphic to $C(Y)$ (i.e. $C(X)\cong C(Y)$) can we deduce that $Y$ is also a zero dimensional space?
(we recall that a topological space $X$ is a zero dimensional space and $C(X)$ is ring isomorphic to $C(Y)$.(i.eif it has a base of clopen subsets.$C(X)\cong C(Y)$) Can we deduce that $Y$ is also Zero dimensional space.
we recall that a topological space $X$ is zero dimensional if it has a base of clopen subsets.
The same Question Canquestion can be Askedasked by Changingexchanging the ring $C(X)$ with the Banach algebra $C_b(X)$ as follows:
$Q_2$:Let $X$ and $Y$ be topological spaces with the previous assumption. If $X$ is a zero dimensional space and $C_b(X)$ is ring isomorphic to $C_b(Y)$.(i.e.$C_b(X)\cong C_b(Y)$) Can we deduce that Y is also Zero dimensional space.
Q2: Let $X$ and $Y$ be topological spaces with the previous assumption. If $X$ is a zero dimensional space and $C_b(X)$ is ring isomorphic to $C_b(Y)$ (i.e. $C_b(X)\cong C_b(Y)$) can we deduce that Y is also a zero dimensional space?
