Negation of countable union of nowhere dense.

Question

According to my references, a topological space is said to be of first category if it can be expressed as countable union of nowhere dense set, where a set is said to be nowhere dense if it's closure has empty interior. A topological space is said to be of second category if it is not of first category. This definition expressed as negation. But I find kind of hard to understand the exact meaning.

I understand Baire's category theorem is probably the answer to this but I do struggle to understand the whole picture.

The question anyway is what's the actual (logical) negation of the definition of first category, if wanted to prove that a set is of second category what exactly should I look for?

Does it mean either uncountable union of nowhere dense or for any sequence of sets whose union gives me the entire space at least the closure of one of them has non empty interior?

Thank you.

Answer 1

The second one is correct. To show that a space is of second category you have to show that if the space is expressed as a countable union of sets then the interior of closure of at least one of then is non-empty. The first one is false.