In a textbook, the following is written.
I have a complete metric space, $S$. I have a union of a sequence of nowhere dense sets. If each set in the sequence is replaced by its closure, then the union will ‘only get larger’.
This appears strange, given that the closure of each of these sets will have empty interior (seeing as the sets are nowhere dense).
I am having some trouble understanding why this is so. Any clarification is much appreciated.