If $P$ and $Q$ are Lebesgue partitions then is $P\cap Q$ a refinement to $P$ and $Q$?

Question

My teacher said that if $P$ and $Q$ are Lebesgue partitions then $P\cap Q$ is a refinement to $P$ and $Q$.

But shouldn't $$P\cup Q$$ be the refinement to both $P$ and $Q$?

Or, are both $P\cup Q$ and $P\cap Q$ refinements to $P$ and $Q$?