Sets that are generic but not of second category and vice versa

Question

I'm reading a chapter on Baire's category theorem in Shakarchi/Stein's Functional analysis. The make the following definitions:

A set is of first category if it is the countable union of nowhere dense subset. A set that is not of first category is of second category.

The complement of a set of the first category is said to be generic.

I wonder if you have any examples of sets that are of the second category but not generic, and vice versa? Is there any criteria for them to coincide or are they totally unrelated?

Kind regards,

Answer 1

In $\mathbb R$ any set with non-empty interior is second category. So $[0,1]$ is second category and so is its complement.

The Baire Category Theorem applies to all complete metric spaces: The complement of a first category set contains a dense $G_{\delta}$ subset.

In order for a set to be generic but not second category the space must be the union of two first category sets, which implies the whole space is a first category set. For example in $\mathbb Q$ let $A=\{\{q\}:q\in \mathbb Q\}.$ Then $\cup A$ is first category and so is its complement $\emptyset.$