Let $D$ be a discrete space of cardinality $\mathfrak{c}$ and $\beta(D)$ is the Stone Cech compactification of $D$. Let $$ X = ((\beta(D) \times (\mathfrak{c}^+ + 1)) \setminus (\beta(D) \setminus D) \times \{\mathfrak{c}^+\}) $$ be the subspace of the product space $\beta(D) \times (\mathfrak{c}^+ + 1)$. Does $X$ has a dense compact subset in $X$, if yes, then what that subset would be.
If not, then here is my another question:
If $\mathcal{U}$ be any family of pairwise disjoint nonempty open sets in $X$, then can we find a compact subset $K$ of $X$ such that $K$ intersects with every member of $\mathcal{U}$.
Any help would be appreciated.