Is it true that $X\times I\sim Y\times I\implies X\sim Y$? So I asked this question a few weeks ago on MSE and I was suggested to repost it here.

Let $I$ be the unit interval. Suppose that $X$ and $Y$ are topological spaces such that $X\times I$ is homeomorphic to $Y\times I$. Does it follow that $X$ is homeomorphic to $Y$?

As pointed out in the comments in the other thread, there are counterexamples to analogous questions with $I$ replaced by the circle or the real line. Therefore I expect the answer to my question to be negative too. I would be also interested in what one can assume about $X$ and $Y$ to make the implication true.
 A: 
@WlodekKuperberg (perhaps) and I (for sure) were exposed to this kind of examples by Karol Borsuk, or possibly Karol Borsuk simply had an example like the one I will present below:

\begin{equation}
D\ :=\ \{z\in\mathbb C: |z|\le 1\}\ \subseteq\ \mathbb C
\
\end{equation}
\begin{equation}
A\,\ :=\,\ D\times\{0\}\ \cup\ \{1\ \ \ i\ \ -\!1\ \ -\!i\}\times [-1;0]\,\ \subseteq\,\ \mathbb C\times\mathbb R
\end{equation}
\begin{equation}
X\,\ :=\,\ A\,\ \cup\,\ \{-1\ \ 1\}\times [0;1]\,\ \subseteq\,\ \mathbb C\times\mathbb R
\end{equation}
\begin{equation}
Y\,\ :=\,\ A\,\ \cup\,\ \{i\ \ \ 1\}\times [0;1]\,\ \subseteq\,\ \mathbb C\times\mathbb R
\end{equation}
Then $\ X\ $ and $\ Y\ $ are not homeomorphic while $\ X\times I\ $
and $\ Y\times I\ $ are.

REMARK One may check Karol Borsuk's series of publications about the uniqueness of topological decomposition into Cartesian products, and a paper by Hanna Patkowska about the uniqueness of the decomposition of ANRs into 1-dimensional ANRs.

A kind request (I'd greatly appreciate): Wlodek Kuperberg, please add a picture to my analytic description; let the pictures of $\ X\ $ and $\ Y\ $ be embedded into $\ \mathbb C;\ $ I mean homeomorphic copies of $\ X\ $ and $\ Y$.

ACKNOWLEDGEMENT I am grateful to Wlodek Kuperberg for providing such a very nice graphics (so cute and psychologically loaded; it's the first graphics illustration in my MO posts)). *** Włodek, congratulation on your another NICE answer (Gauss said, a few but ripe).
A: There are indeed counterexamples to which Igor Belegradek gave reference. Here is another counterexample in the plane, perhaps the simplest there is: Let $X$ be an annulus with one arc attached to one of its boundary components and another arc attached to the other boundary component, and $Y$ -  an annulus with two disjoint arcs attached to the same one of its boundary components.

