We say that a space $(X,\tau)$ has the *fixed point property (FPP)* if for every continuous map $f:X\to X$ there is $x\in X$ with $f(x) = x$.


What is an example of a space $X$ with FPP such that $X^2$ (with the product topology) does not have FPP?



Squaring a space with the fixed-point property
The topological space
$$
X = \{ (x,\sin \left( \frac{\pi}{1-x} \right)) \ | \ 0 \leq x < 1 \} \cup {(1,1)} \subseteq \mathbb{R}^2
$$
has the FPP but its square does not. See Example $2$ p.$977$ of [E. H. Connell, Properties of Fixed Point Spaces](https://www.jstor.org/stable/pdf/2033633.pdf?refreqid=excelsior%3A7894cfc6d765c45b212d8996d81bc6e5).


