Decompostion into countably many nowhere dense compact sets

93 Views Asked by At

Let $A$ be a meagre subset of a locally compact abelian Polish group $G$. Then $A$ can be written as a countable union of nowhere dense subsets of $G$.

Is it always possible to write $A$ as a countable union of nowhere dense compact subsets of $G$ ?

1

There are 1 best solutions below

3
On BEST ANSWER

No. Take an uncountable subset $S$ of the Cantor ternary set which does not contain any perfect set. Then $S$ is meager (even nowhere dense) in the real line but every compact subset of $S$ is countable so it cannot be covered by countably many compact subsets.