Subset of Discrete Set of Subsets is Discrete

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Theorem

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

Let $\FF \subseteq \powerset S$ be a set of subsets of $S$.

Let $\GG \subseteq \FF$.

If $\FF$ is discrete then $\GG$ is discrete.

Proof

We prove the contrapositive statement:

If $\GG$ is not discrete then $\FF$ is not discrete.

Let $\GG$ not be discrete.

By definition of discrete:

$\exists x \in S : \forall N \subseteq S : N$ is a neighborhood of $x : \exists X_1, X_2 \in \GG : X_1 \ne X_2: X_1 \cap N \ne \O, X_2 \cap N \ne \O$

By definition of subset:

$\forall X \in \GG : X \in \FF$

Hence:

$\exists x \in S : \forall N \subseteq S : N$ is a neighborhood of $x : \exists X_1, X_2 \in \FF : X_1 \ne X_2: X_1 \cap N \ne \O, X_2 \cap N \ne \O$

By definition, $\FF$ is not discrete.

The result follows from Rule of Transposition.

$\blacksquare$