Union of Meager Sets is Meager Set

Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $A$ and $B$ be meager in $T$.

Then $A \cup B$ is meager in $T$.

Proof

Since $A$ is meager in $T$:

there exists a countable collection of sets $\set {U_n: n \in \N}$ nowhere dense in $T$ such that $\ds A = \bigcup_{n \mathop \in \N} U_n$.

Since $B$ is meager in $T$:

there exists a countable collection of sets $\set {V_m: m \in \N}$ nowhere dense in $T$ such that $\ds B = \bigcup_{m \mathop \in \N} V_m$.

Then:

$\ds A \cup B = \paren {\bigcup_{n \mathop \in \N} U_n} \cup \paren {\bigcup_{m \mathop \in \N} V_m}$

The right hand side is a countable union of sets nowhere dense in $T$, so:

$A \cup B$ is meager in $T$.

$\blacksquare$