Components of Separation are Separated Sets
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Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $A \mid B$ be a separation of $T$.
Then $A$ and $B$ are separated sets of $T$.
Proof
By definition of closure, $A^-$ is the smallest closed set of $T$ that contains $A$.
Components of Separation are Clopen shows that $A$ and $B$ are closed.
It follows that $A^- = A$, and $B^- = B$.
Definition of separation shows that $A \cap B = \O$, so we have:
| \(\ds A^- \cap B\) | \(=\) | \(\ds A \cap B\) | \(\ds = \O\) | |||||||||||
| \(\ds A \cap B^-\) | \(=\) | \(\ds A \cap B\) | \(\ds = \O\) |
Hence the result by definition of separated sets.
$\blacksquare$