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 collection of sets $\set {U_\alpha: \alpha \in X}$ nowhere dense in $T$ such that $\ds A = \bigcup_{\alpha \in X} U_\alpha$.

Since $B$ is meager in $T$:


 * there exists a collection of sets $\set {V_\beta: \beta \in Y}$ nowhere dense in $T$ such that $\ds B = \bigcup_{\beta \in Y} V_\beta$.

Then:


 * $\ds A \cup B = \paren {\bigcup_{\alpha \in X} U_\alpha} \cup \paren {\bigcup_{\beta \in Y} V_\beta}$

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


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