Components of Separation are Clopen

Theorem

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $A \mid B$ be a separation of $T$.

Then both $A$ and $B$ are clopen in $T$.

Proof

From Set with Relative Complement forms Partition:

$A = \complement_S \left({B}\right)$

and:

$B = \complement_S \left({A}\right)$

where $\complement_S$ denotes the complement relative to $S$.

As $A$ and $B$ are both open, it follows by definition that $A$ and $B$ are also both closed.

That is, by definition, they are clopen.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $6.2$: Connectedness: Definition $6.2.2$