State a version of the Baire category theorem for a complete metric space. Let $T$ be the set of real numbers $x$ with the property that, for each positive integer $n$, there exist integers $p$ and $q$ with $q \geqslant 2$ such that

$$0<\left|x-\frac{p}{q}\right|<\frac{1}{q^{n}} .$$

Is $T$ an open subset of $\mathbb{R}$ ? Is $T$ a dense subset of $\mathbb{R}$ ? Justify your answers.