The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise true that the countable intersection of dense open subsets of a complete uniform space is dense?
A look at the proof suggests that the answer should be no—it makes crucial use of the fact that the canonical uniform structure on a metric space is generated by the $\varepsilon$-balls of radius $\frac{1}{m}$ for $m\ \in \mathbb{Z}^+$. I thus started looking at complete topological groups that were not first-countable for a counter-example, with no luck (topological groups because they provide easy examples of uniform spaces, and not first-countable because first-countable topological groups have uniformities generated by a countable collection of covers (and also because (I think?) first-countable topological groups are metrizable).)
Surprisingly, a quick Google search only pulled up one source that seems to mention this at all—Joshi in his Introduction to General Topology states (pg. 363)
Unfortunately, there seems to be no analogue of the Baire category theorem for complete uniform spaces.
but yet does not provide a counter-example either.
Also, I doubt this will make too much of a difference, but I would prefer the counter-example to be $T_0$. Finding an example of a highly pathological space that is, uhh, pathological, isn't exactly what I'm looking for.
Best Answer
Functional analysis abounds with such examples. There are many complete locally convex spaces which are countable unions of closed subspaces with empty interior. One such is the space of smooth functions on the line which have compact support, the test functions of L. Schwartz. An even simpler one is the space of finite sequences with the locally convex inductive limit topology as the union of finite dimensional spaces.
Added as an edit: The completeness of the first space follows from the fact that it is a strict $LF$-space and these are complete by a result of Dieudonné and Schwartz, of the second from the fact that it is the strong dual of the (nuclear) Fréchet space of all sequences.