Finding a space with $X \cong X+2$ and $X \not\cong X+1$.

Question

Question. Is there a topological space $X$ with $X \cong X+2$ and $X \not\cong X+1$?

Here, $X+n$ denotes the disjoint union (i.e. coproduct) of $X$ with $n$ isolated points.

This question is similar to MO/218113 and MO/225896. I am pretty sure that it is easier, though. Perhaps it already works with a nasty topology on $\mathbb{N}$?

Answer 1

A reference to such a space and a brief description can be found in this answer; there is a more thorough description in this answer. Briefly, the space is obtained by taking two copies of $\beta\Bbb N$, the Čech-Stone compactification of $\Bbb N$, and identifying the remainders in the obvious way.