Closure of Complement of Closure is Regular Closed

Theorem

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

Let $A \subseteq S$ be a subset of $T$.

Let $A^-$ denote the closure of $A$ in $T$.

Let $A'$ denote the complement of $A$ in $S$: $A' = S \setminus A$.

Then $A^{-'-}$ is regular closed.

Proof

Let $A^\circ$ denote the interior of $A$.

From Set is Closed iff Equals Topological Closure, $A^-$ is closed in $T$.

Since $A^-$ is a closed set, $A^{-'}$ is open.

Therefore:

$\forall x \in A^{-'}: \exists \epsilon \in \R_{>0}$

prime.mover suggests: The validity of the material on this page is questionable. In particular: The above statement makes no sense in this context You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Additionally, because $A^{-'}$ is the complement of $A^-$ relative to $S$, we have:

$\exists \epsilon \in \R_{>0}: \map {V_\epsilon} x \cap A^- = \O \iff x \in A^{-'}$

prime.mover suggests: The validity of the material on this page is questionable. In particular: The above $\map {V_\epsilon} x$ cannot be applied in the context of a general topological space. If it were stated that $T$ were a metric space, then maybe (with some further work to establish links to results to back the statement up), but it isn't so it can't. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Questionable}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Therefore, for $y: y \in A^{-'-}$ and $ y \notin A^{- '}$ we know that:

$\forall \epsilon \in \R_{>0}: \map {V_\epsilon} y \cap A^- \ne \O$

meaning that $y$ is a limit point of $A^-$.

Therefore:

$y \notin A^{-'-\circ}$

Now:

$x \in A^{-'-\circ} \iff x \in A^{-'}$

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In conclusion, this means that:

$A^{-'-\circ-} = A^{-'-},$ verifying the definition of $A^{-'-}$ being regular closed.

$\blacksquare$