Set of Condensation Points of Union is Union of Sets of Condensation Points
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Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.
Let $A, B$ be subsets of $S$.
Then:
- $\left({A \cup B}\right)^0 = A^0 \cup B^0$
Lemma
Let $x$ be a point of $S$.
Then:
- if $x$ is condensation point of $A \cup B$,
- then $x$ is condensation point of $A$ or $x$ is condensation point of $B$.
Proof
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:
- $x \in A^0 \cup B^0$
$\Box$
Union of Sets of Condensation Points Subset Set of Condensation Points 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:
- $A^0 \cup B^0 \subseteq \left({A \cup B}\right)^0$
$\Box$
Thus by definition of set equality:
- $\left({A \cup B}\right)^0 = A^0 \cup B^0$
$\blacksquare$
Sources
- Mizar article TOPGEN_4:54