In a lecture on Baire's Category Theorem at Indian Institute of Tech, it was mentioned that the converse of Baire's: Every non meagre (second category) space is complete, is not true, and that a proof of the existence of an incomplete non meagre (second category) space was given by Bourbaki.
(N.B. The course used a weaker form of Baire's than usual: Every complete space is non meagre (of second category).)
Question: First of all I have failed to find the proof mentioned, but would also like to ask if anyone can and would give me a short version of why this is, and also tell me what's wrong (the existence of a known proof suggests there is) with the following trivial counterexample:
The interval $(0,1)$ is incomplete as a metric subspace of $\mathbb{R}$, yet it is non meagre.
Any open subset of a complete metric space (more generally and $G_{\delta}$ subset) has an equivalent metric which makes it complete. So it is non-meager.
In the case of $(0,1)$ such a metric is defined by $D(x,y)=|x-y|+|\frac 1 {d(x)} -\frac 1{d(y)}|$ where $d(x) =\min \{{x, 1-x}\}$.