Counterexample to the Converse of Baire's Category Theorem

Solution 1:

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 $d(x) =\min \{{x, 1-x}\}$.

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Counterexample to the Converse of Baire's Category Theorem

Solution 1:

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