If $P$ is perfect and $K $ is compact then what can we say that $P\bigcup K$

Question

If $P$ is perfect and $K $ is compact then what can we sat that $P\bigcup K$

Set $P$ is perfect $\rightarrow$ closed and contains no isolated points

Set $K$ is compact $\rightarrow$ closed and bounded

I know that $P\bigcap K$ is compact.

Answer 1

Union of two closed sets is closed.

Answer 2

We can only say $P\cup K$ is closed.

$P\cup K$ may not be compact as $P=\Bbb{R}$ is a counter example.

$P\cup K$ may not be perfect either as any $K$ that $K-P$ is not perfect is counter example.