If a closed set is the union of two disjoint sets and one set is open, is the other set closed/clopen/neither?

Question

Suppose we have set $X = A \cup B$ where $A, B$ are disjoint, $X$ is closed and $A$ is open.

$B$ cannot be open since this would imply $X$ is open (arbitrary unions of open sets is open), which is a contradiction.

Does this imply $B$ is closed, or could it be neither?

Answer 1

$B$ is closed because $B=X\setminus A=X\cap A^C$ and both $X$ and $A^C$ (the complement of $A$) are closed.

On top of that, $B$ may or may not be open. If every set is open (discrete topology) then $B$ is open too. On the other hand, take $X=B[0,2]$ (closed ball with radius $2$) in $\mathbb R^2$ and take $A=B(0,1)$ (open ball with radius $1$): with $B=X\setminus A$ you can see that the resulting "ring-shaped" $B$ is closed but not open.

Answer 2

Hint Since the sets are disjoint, $B$ is the intersection between the closed sets $X$ and the complement of $A$.

Note that in the discrete topology, all three sets could be both closed and open.