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

Then, we have:

From Intersection is Subset:


 * $U_n \cap B \subseteq U_n$

From Subset of Nowhere Dense Subset is Nowhere Dense:


 * $U_n \cap B$ 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 $T$.