Suppose there are compact sets $K_1,K_2\subset\mathbb{R}^n$ such that $K_1\times \mathbb{R}\cong K_2\times \mathbb{R}$,
but $K_1\ncong K_2$.

>What is the minimum of $n$?

**Comments**

* The spherical suspension over the Poincaré homology sphere and $\mathbb{S}^4$ provide an example for $n=5$.

* If instead of $\mathbb{R}$ one takes the closed interval $[0,1]$, then examples of planar compact sets are shown below. See also [this question][1].

     [![enter image description here][2]][2]

* If one removes the assumption of compactness, then examples can be constructed in a similar way (for $n=2$).


  [1]: https://mathoverflow.net/q/26385
  [2]: https://i.sstatic.net/y4o2Kh0w.png