Let $X$ be a Baire space.
A subset $E\subset X$ is said to be of first category if it can be expressed as the union of countably many nowhere dense subsets of $X$. Then $E$ is said to be of second category if it is not of first category.
A subset $E\subset X$ is said to be residual if its complement $X\backslash E$ is of first category.
Then $E\subset X$ is said to be locally residual if there exists a nonempty open subset $U\subset X$ such that $E\cap U$ is residual in $U$.
My question is: if a subset is of second category, does it have to be locally residual?
There might be some counterexamples. I have no idea. Thank you!