Subset of Meager Set is Meager Set

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

Then, we have:

\(\ds B\) \(=\) \(\ds A \cap B\) Intersection with Subset is Subset
\(\ds \) \(=\) \(\ds \paren {\bigcup_{n \mathop \in \N} U_n} \cap B\)
\(\ds \) \(=\) \(\ds \bigcup_{n \mathop \in \N} \paren {U_n \cap B}\) Union Distributes over Intersection

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 $T$.

$\blacksquare$