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Community Bot
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In general the answer to both questions is no: there are pseudocompact zero-dimensional spaces (see, e.g., this answersee, e.g., this answer) whose Cech-Stone compactifications are not zero-dimensional. For an $X$ like that one has $C(X)=C_b(X)$ and this ring is isomorphic to $C(\beta X)$.

In general the answer to both questions is no: there are pseudocompact zero-dimensional spaces (see, e.g., this answer) whose Cech-Stone compactifications are not zero-dimensional. For an $X$ like that one has $C(X)=C_b(X)$ and this ring is isomorphic to $C(\beta X)$.

In general the answer to both questions is no: there are pseudocompact zero-dimensional spaces (see, e.g., this answer) whose Cech-Stone compactifications are not zero-dimensional. For an $X$ like that one has $C(X)=C_b(X)$ and this ring is isomorphic to $C(\beta X)$.

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KP Hart
KP Hart
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In general the answer to both questions is no: there are pseudocompact zero-dimensional spaces (see, e.g., this answer) whose Cech-Stone compactifications are not zero-dimensional. For an $X$ like that one has $C(X)=C_b(X)$ and this ring is isomorphic to $C(\beta X)$.