Example of non-homeomorphic compact spaces $K_1$ and $K_2$ such that $K_1\oplus K_1$ is homeomorphic to $K_2\oplus K_2$

Solution 1:

K. Sundaresan, Banach spaces with Banach-Stone property, in Studies in Topology (N.M. Stavrakas & K.R. Allen, eds.), Academic Press, New York, $1975$, pp. $573$-$580$, contains an example of a compact Hausdorff space $X$ such that if $Y$ and $Z$ are the results of adding one and two isolated points, respectively, to $X$, then $X\cong Z\not\cong Y$, where $\cong$ denotes homeomorphism. Thus, $X\oplus X\cong X\oplus Z\cong Y\oplus Y$, even though $X\not\cong Y$. In On an example of Sundaresan, Topology Proceedings, Vol. $5$ ($1980$), pp. $185$-$186$, I gave a simpler proof of some of the properties of $X$; this paper is freely available here [PDF].

Briefly, $X$ is obtained by pasting together the remainders of two copies of $\beta\omega$ in the natural way. Let $X=\omega^*\cup(\omega\times 2)$, where $\omega^*=\beta\omega\setminus\omega$, and let $\pi:X\to\beta\omega$ be the obvious projection; the topology on $X$ is the coarsest making $\pi$ continuous and each point of $N=\omega\times 2$ isolated. For $i\in 2$ let $N_i=\omega\times\{i\}$. Intuitively $Y$, obtained by adding an isolated point to $X$, is not homeomorphic to $X$ because the extra point must be added to one of the ‘tails’ $N_0$ and $N_1$, and this ‘skews’ the pasting-together of the two copies of $\omega^*$ to form $X$; in $Z$, on the other hand, we can think of one of the new points as extending $N_0$ and the other, $N_1$, so that the two copies of $\omega^*$, being similarly ‘shifted’, still line up correctly. The actual argument can be found in the linked paper.