Let $K$ be a compact subset of an arbitrary open set $\Omega\subset \mathbb{R}^n.$ It is said that a connected component $W$ of $\Omega\setminus K$ is $\Omega$-bounded if $\overline{W}$ is a compact subset of $\Omega$.

Heuristically speaking, the $\Omega$-bounded components of $\Omega\setminus K$ are the "holes" of $K$ that do not contain "holes" of $\Omega$. Now, I want to prove this, but I am lost. Topology is not my usual work field.

Specifically, I want to prove that if $W$ is an $\Omega$-bounded component of $\Omega\setminus K$ and $H$ is the component of $\mathbb{R}^n\setminus K$ that contains $W,$ then $H\subset \Omega.$ Additionally, is it possible to prove that $H=W$?