Baire's Category Theorem fails for uncountable intersection of open, dense subsets

Question

Since $\mathbb{Z}$ is not dense in $\mathbb{R}$, but equals $\cap_{x\in\mathbb{R}\cap\mathbb{Z}^c}[(-\infty,x)\cup(x,\infty)]$, where each $[(-\infty,x)\cup(x,\infty)]$ is dense and open in $\mathbb{R}$, can I use this to show the necessity of the intersection being countable in the Baire Category Theorem?

Answer 1

Any set of reals whatsoever is the intersection of a family of dense open sets: for $x\not\in A\subseteq\mathbb{R}$, let $D_x=\mathbb{R}\setminus \{x\}$; this is dense and open, and clearly we have $$A=\bigcap_{x\not\in A} D_x.$$ So it is absolutely essential to the Baire Category Theorem that the collection of dense open sets in question be countable.

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EDIT: it may be more intuitive to think of the converse task - expressing any set of reals as a union of closed nowhere-dense sets. The idea is identical, but it might be clearer: we always have $$A=\bigcup_{a\in A}\{a\},$$ and every singleton $\{a\}$ is closed and nowhere dense, so any set of reals is a union of closed nowhere-dense sets.

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EDIT $2$: In response to your most recent edit, yes, writing (for example) $\mathbb{Z}$ as an intersection of open dense sets shows that countability is necessary. An even better example though: what is $$\bigcap_{r\in\mathbb{R}}(\mathbb{R}\setminus \{r\})?$$