Set of Condensation Points of Union is Union of Sets of Condensation Points

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $A, B$ be subsets of $S$.

Then:
 * $\left({A \cup B}\right)^0 = A^0 \cup B^0$

Set of Condensation Points of Union Subset Union of Sets of Condensation Points
Let $x \in \left({A \cup B}\right)^0$.

By definition of set of condensation points:
 * $x$ is condensation point of $A \cup B$

By Lemma:
 * $x$ is condensation point of $A$ or $x$ is condensation point of $B$

By definition of set of condensation points:
 * $x \in A^0$ or $x \in B^0$

Thus by definition of union:
 * $x \in A^0 \cup B^0$

Union of Sets of Condensation Points Subset Set of Condensation Points of Union
By Set is Subset of Union:
 * $A \subseteq A \cup B \land B \subseteq A \cup B$

By Set of Condensation Points is Monotone:
 * $A^0 \subseteq \left({A \cup B}\right)^0 \land B^0 \subseteq \left({A \cup B}\right)^0$

Thus by Union is Smallest Superset:
 * $A^0 \cup B^0 \subseteq \left({A \cup B}\right)^0$

Thus by definition of set equality:
 * $\left({A \cup B}\right)^0 = A^0 \cup B^0$