# Corollary to Baire's Category Theorem

In Rudin's Real and Complex Analysis (p. 97 in my 3rd edition), the following is stated as a corollary to Baire's Category Theorem:

"In a complete metric space, the intersection of any countable collection of dense $G_{\delta}$'s is again a dense $G_{\delta}$."

Proof: "This follows from the theorem, since every $G_{\delta}$ is the intersection of a countable collection of open sets, and since the union of countably many countable sets is countable."

I don't understand the word union in the argument. It seems like it should be intersection, as the theorem states that such an intersection is dense. Is this a typo or am I missing something?

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Union (as in set union) is the correct word. Assuming the theorem, we just need to show that a countable collection of $G_\delta$'s is again countable, therefore satisfying the assumptions of the theorem. –  jericson Jan 4 '11 at 23:11

"Union" is the correct word here. The point is that we have an iterated intersection: we have for each $i$ a countable family $\{U_{ij}\}$ of dense open subsets and we are taking the intersection over all $i$: $\bigcap_{i \in I} \bigcap_{j \in J} U_{ij}$.
Here $I$ and $J$ are both countable index sets. This intersection can be written as $\bigcap_{(i,j) \in I \times J} U_{ij}$. The passage you quote is giving an argument that $I \times J$ is again a countable index set: it is the countable union of the countable sets $\{i\} \times J$ as $i$ ranges over all elements of $I$. (One could give other arguments for this as well.)
Yes. $\textbf{}$ –  Pete L. Clark Jan 5 '11 at 0:19