Does there exist a second category set that is not a Baire space?

Question

Can someone give an example of a second-category set $Y$ in a metric space $X$ but $Y$ is not a Baire space.

We know that Baire space$\implies $ Second Category.

But I am trying to show that the converse in not true.

Clearly ,$Y$ must not be complete.

Answer 1

Let $X=\Bbb R$ with the usual topology, and let $Y=[0,1]\cup(\Bbb Q\cap[2,3])$; $Y$ is second category in $X$, but it’s not a Baire space, since the sets $Y\setminus\{q\}$ for $q\in\Bbb Q\cap[2,3]$ are dense and open in $Y$, but their intersection $[0,1]$ is not dense in $Y$.