I have a complete metric space $Y$, some non-metrizable(!) Hausdorff compactification $Z$ of it and a subspace $X \subset Y$.


Furthermore, I do have a uniformly continuous function $f$ on $X$. So there is a uniformly continuous extension of $f$ from $X$ to the closure of $X$ in $Y$.



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> Can we extend f to a uniformly continuous function on the closure of $X$ in $Z$?
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For this we equip $Z$ with the unique uniform structure which all compact Hausdorff spaces posses.


It seems that this question boils down to the question whether the from $Z$ induced uniform structure on $X$ coincides with the one coming from the metric on $X$. But sadly, I can't answer this on my own since I'm not familiar with uniform spaces.


If the answer does depend on $Z$, I'm interested in $Z = \beta Y$ (the Stone-Cech compactification).



Extending uniformly continuous functions on subspaces to non-metrizable compactifications
The closure of $X$ in $Z$ is compact , so there is no hope if $f$ is not bounded. If it is bounded then so is its extension to the closure of $X$ in $Y$ and this gives a bounded uniformly continuous function on the closure which can be extended to a bounded continuous function on $Y$ by the Tietze extension theorem. This, in turn, extends to a continuous function on the Stone-čech compactification. You can then restrict the latter to the closure of $X$ in $Z$.


