We know that the space $\mathbb{R}$ has compactifications with one point remainder, and two point remainder. but there is no compactification of $\mathbb{R}$ with three point remainder and the same holds for every finite natural number $n$ greater than 3. We know that the Stone-Čech compactification of $\mathbb{R}$ has infinite remainder. (i.e.$|\beta\mathbb{R}-\mathbb{R}|=2^\mathfrak{c}$
I have the same question for other infinite cardinals less than $2^\mathfrak{c}$ as follows:
A. Is there any compactification $X$ of $\mathbb{R}$ with the property that $|X-\mathbb{R}|=\aleph_0$?
B. Can we improve our question to cardinals less than $2^{\aleph_0}$?