Union of Set of Dense-in-itself Sets is Dense-in-itself
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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
- Mizar article TOPGEN_1:38