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Let $X$ be a completely regular space and let $T$ a topological space such that $X \subseteq T \subseteq \beta X$. Then $\beta T = \beta X$, where $\beta$ denotes the Stone-Cech compactification.

Solution: Let $f: T \mapsto [0,1]$. Then it suffices to show that the restriction of f to $X$ can be extended to $\beta X$"..

Why is this?

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    $X$ doesn't seem to be used here. Couldn't you just say you have $C\subseteq D\subseteq Z$?2011-01-17
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    It doesn't. Are you sure there are no additional conditions on $X$? (say, is it Hausdorff? Regular?) or on $f$? or on $C$ and $D$? (say, is $C$ dense in $D$?)2011-01-17
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    I guess if $X$ is normal then we could use the following: http://en.wikipedia.org/wiki/Tietze_extension_theorem2011-01-17
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    @Andres Caicedo: Just corrected it. The original post was a mess, I am sorry.2011-01-17

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