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 $\ds A = \bigcup_{n \in \N} U_n$.

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 $\ds B = \bigcup_{m \in \N} V_m$.

Then:


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

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


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