I would like to know if the extension theorem of uniformly continuous functions can be generalized to pseudometric spaces.
That is, let $X,Y$ be a pseudometric spaces and $D\subset X$ a dense subset. If $f:D\rightarrow Y$ is uniformly continuous then can it be extended uniquely to a continuous function?
Uniform continuity can be defined analogously to the metric case. I'm not sure the same proof works because we don't have continuity iff preservation of convergence.
I guess we may need completeness of $Y.$