Determine category of the sets

Question

Determine category of the sets

$A = \{1\}, B=\{0\}, C=\bigcup_{n=1}^\infty \{1/n\},$ in the space $\mathbb{R}$ with $d(x,y) =|x|+|y|$ for $x\neq y.$

In usual metric I know that singletons are nonmeager since they cannot be split but I confused when metric changed to $d(x,y) =|x|+|y|$.

Answer 1

With this metric, $\{x\}$ is open-and-closed for every $x\in \Bbb R$ \ $\{0\}$ because $B_d(x,|x|/2)=\{x\}$ if $x\ne 0.$

So if $x\ne 0$ then $\{x\}$ is not meager. Because if $\{x\}=\cup_{n\in \Bbb N}S_n$ then $S_n=\{x\}$ for some $n,$ whereupon $int(S_n)=S_n =\{x\} \ne \phi$, so $S_n$ is NOT nowhere dense.

Similarly, $C$ is a countably infinite open set so if $C=\cup_{n\in \Bbb N}T_n$ then some (any) non-empty $T_n$ has non-empty interior (because any non-empty subset of $C$ is open). So $C$ is not meager.

On the other hand $int(\{0\}$ is empty. Because if $r>0$ then $B_d(0,r)=\{y\in \Bbb R: |y|<r\}$ is not a subset of $\{0\}.$ But since $\{0\}$ is also closed, $\{0\}$ is therefore a closed nowhere-dense set, hence it is meager.