3 edited title

2 edited tags
1

Existence of a compactification of $\mathbb{R}$ with $\aleph_0$ reminder

We now that the space $\mathbb{R}$ has compactifications with one point reminder, and two point reminder. but there is no compactification of $\mathbb{R}$ with three point reminder and the same holds for every finite natural number $n$ greater than3. We Know that the stone-cech compactification of $\mathbb{R}$ has infinite reminder. (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 or question to cardinals less than $2^{\aleph_0}$?