Meager, Nonmeager, and Comeager

Question

A set is meager if it is countable union of nowhere dense sets and otherwise it is nonmeager. A set is comeager if it is complement of meager set. I have two problems. Let $X$ be a topological space.

1. With this definition, is it true that any set $A \subseteq X$ is meager or nonmeager?
2. I can not find $A \subseteq X$ which is not meager and not comeager. Can you help me to find this set? Thank you.

Answer 1

1. excluded middle!

2. For instance $X=\Bbb R$ and $A=\{x:x>0\}$.