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2 | corrected a typo---$X$ for $Y$. | ||
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Question 1 has a positive answer for general $X$. The reason for this is that $C(X)$ cannot distinguish between $X$ and its realcompactification. Hence if $X$ is not realcompact we construct $M_{x,y}$ as above with $x$ in $Y$ X$ and with $y$ in the realcompactification, but not in $X$. This is a maximal subring which does not have the required form. A good reference for this material is Weir's book "Hewitt-Nachbin spaces" (and, of course, Gilman and Jerison). |
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Question 1 has a positive answer for general $X$. The reason for this is that $C(X)$ cannot distinguish between $X$ and its realcompactification. Hence if $X$ is not realcompact we construct $M_{x,y}$ as above with $x$ in $Y$ and with $y$ in the realcompactification, but not in $X$. This is a maximal subring which does not have the required form. A good reference for this material is Weir's book "Hewitt-Nachbin spaces" (and, of course, Gilman and Jerison). |
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