Suppose that $A$ is of the first category. Is it possible to write $A$ as a countable union of sets that are not necessarily nowhere dense? I mean after all a first category set is defined such that it can be expressed as a countable union of nowhere dense sets.
I am a bit confused by the definition of first category sets...If I can find a representation of $A$ such that it is the countable union of sets, some of which are not nowhere dense, is it still possible that $A$ is a first category set?
A first category set $A$ is a set that can be written as a countable union of nowhere dense sets. However, that might not be the only way to write it as a countable union.
Indeed, there is a very trivial way to write any set at all as a countable union of sets. Think about the most trivial way to do this.
Now recall any first category set that isn't nowhere dense, and see how you can write it as a counterexample.