Union of Derivatives is Subset of Derivative of Union

Theorem

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

Let:

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

where $\powerset S$ denotes the power set of $S$.

Then:

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

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

Proof

Let $\ds x \in \bigcup_{A \mathop \in \FF} A'$.

Then by definition of union there exists $A \in \FF$ such that:

$(1): \quad x \in A'$

By Set is Subset of Union:

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

Then by Derivative of Subset is Subset of Derivative:

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

Hence by $(1)$ the result:

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

follows by definition of subset.

$\blacksquare$

Sources

• 1977: Ryszard Engelking: General Topology

• Mizar article TOPGEN_1:34