Subset of Meager Set is Meager Set

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

Let $A$ be meager in $T$.

Let $B \subseteq A$.

Then $B$ 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 $\ds A = \bigcup_{\alpha \in X} U_\alpha$.

Then, we have:

From Intersection is Subset:


 * $U_\alpha \cap B \subseteq U_\alpha$

From Subset of Nowhere Dense Subset is Nowhere Dense:


 * $U_\alpha \cap B$ is nowhere dense in $T$.

Then, we see that:


 * $B$ can be written as the union of nowhere dense sets in $T$.

That is:


 * $B$ is meager in $T$.