# Components of Separation are Separated Sets

Jump to navigation
Jump to search

## 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.

$\blacksquare$