Union of Set of Dense-in-itself Sets is Dense-in-itself

Theorem

Let $T$ be a topological space.

Let $\FF \subseteq \powerset T$ such that

every element of $\FF$ is dense-in-itself.

Then the union $\bigcup \FF$ is also dense-in-itself.

Proof

By Dense-in-itself iff Subset of Derivative:

$\forall A \in \FF: A \subseteq A'$

where $A'$ denotes the derivative of $A$.

Then by Set Union Preserves Subsets:

$\ds \bigcup \FF \subseteq \bigcup_{A \mathop \in \FF} A'$

By Union of Derivatives is Subset of Derivative of Union:

$\ds \bigcup_{A \mathop \in \FF} A' \subseteq \paren {\bigcup \FF}'$

Then by Subset Relation is Transitive:

$\ds \bigcup \FF \subseteq \paren {\bigcup \FF}'$

The result follows by Dense-in-itself iff Subset of Derivative.

$\blacksquare$

Sources

• 1989: Ryszard Engelking: General Topology (revised and completed ed.)

• Mizar article TOPGEN_1:38