I am trying to prove Kuratowski Complement-Closure Theorem, which states that if we start with a set $S \subseteq X$ in a topological space $(X, \tau)$, we can get at most $14$ distinct sets using the operation of taking complements and closures. I am trying to prove the following fact first: if we start with a closed set, then the most number of sets we can form with these two operations is $6$. Here is a write-up of the proof.
We now show that $B, B ^ C, \overline{B ^ C}, \overline{B ^ C} ^ C, \overline{\overline{B ^ C} ^ C},$ and $\overline{\overline{B ^ C} ^ C} ^ C$ are the only distinct sets that we can form. Let $F$ denote $\overline{\overline{B ^ C} ^ C} ^ C$. We now show that $\overline F = \overline{B ^ C}$. We see that $$B \ is \ closed \Rightarrow B ^ C = Int(B ^ C)$$ $$ \Rightarrow \overline{B ^ C} = B ^ C \cup Bd(B ^ C) $$ $$\Rightarrow \overline{B ^ C} ^ C = B \cap Bd(B ^ C) ^ C $$ $$= B \cap Bd(B) ^ C $$ $$= Int B \cup Bd(B) \cap Bd(B) ^ C = Int B$$ $$ \Rightarrow \overline{\overline{B ^ C} ^ C} = \overline {Int(B)} $$ $$\Rightarrow F = \overline{Int(B)} ^ C = Int(B) ^ C - Bd(Int B)$$.
Hence, we see that $x \in \overline F \Rightarrow $every neighborhood of $x$ intersects $Int(B) ^ C - Bd(Int B)$. We now show that $\overline{B ^ C} \subseteq \overline F$. We only need to show that $B ^ C \subseteq \overline F$. If $x \in B ^ C$, then $x \in Int(B ^ C)$, which means that there exists a neighborhood of $x$ that is contained in $B ^ C$ (as $B ^ C$ is open). Therefore, we see clearly that $x \not \in Bd(Int B)$ and $x \in Int(B) ^ C$. Hence, we see that $B ^ C \subseteq \bar F$. We now show that $F \subseteq \overline{B ^ C}$. If $x \in Int (B) ^ C - Bd(Int B)$, we then see that there exists a neighborhood of $x$ that does not intersect $Int(B)$ and $x \not \in Int(B)$. If there exists a neighborhood of $x$ that only intersects $B$, then clearly $x \in Int(B)$, which is a contradiction. Therefore, we see that $x \in \overline{B ^ C}$, which means that $\overline F = \overline{B ^ C}$.
I was wondering if anyone could check to see if there is any error in the proof. Thanks in advance!