Let $(G_k)$ a sequence of open and dense subsets. Show that $\bigcap_{k\geqslant 0}G_k$ is also open and dense

Question

The exercise says

Let $(G_k)$ a sequence of open and dense subsets in a complete metric space. Show that $\bigcap_{k\geqslant 0}G_k$ is also open and dense

I have shown that $G:=\bigcap_{k\geqslant 0}G_k$ is dense, however I think that I have a counterexample for $G$ to be also open. If it would be true then it must also be true that $G^\complement =\bigcup_{k\geqslant 0}G_k^\complement $ would be closed and with empty interior. However setting $G_k^\complement :=\{1/k\}$ we have that $G^\complement=\{1/k:k\in \Bbb N_{> 0} \} $ have empty interior in $\Bbb R$ but it is not closed because zero is a limit point of $G^\complement $.

Im wrong or the exercise is wrong?

Answer 1

The exercise is wrong. The intersection of countably many open dense sets is dense, as you know, but doesn't have to be open. It is by definition $G_\delta$, the intersection of countably many open sets. Another example is if $G_k=\mathbb{R}\setminus\{{r_k}\}$ where $r_1,r_2,\ldots$ is an enumeration of the rationals. Then the intersection of the $G_k$ is the irrational numbers, which is definitely dense and not open.

Answer 2

That error was in the manuscript for my forthcoming book Measure, Integration & Real Analysis, which will be published in December 2019 in Springer's Graduate Texts in Mathematics series. I meant for the exercise to state that the intersection of a sequence of dense open subsets of a complete metric space is dense, but a copy-and-paste error left "open" also in the conclusion. This error has been corrected in the most recent version of the manuscript, which is freely available on the book's website (http://measure.axler.net).

This error has also been corrected for the print version of the book. Because this book will be published in Springer's Open Access program, the electronic version of the book will remain free to the world even after the print version is published.