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Let $X$ be a compact metric space with topological dimension ${\rm dim}(X)>0$. Let $f: X\times X\to X$ be a continuous and surjective map. Define a family of maps $f_n: X^{n+1}\to X^{n}$ for $n\ge 1$ by $f_1=f$ and $f_{n+1}(x_1, x_2, \dots, x_{n+1})=(f_{n}(x_1, \dots, x_{n}), f(x_n, x_{n+1}))$. I think that the topological dimension of the inverse limit of the family $(X^n, f_n)$ is infinite. But I can not prove it. I will be appreciated if anyone could prove such statement or disprove by a counter-example.

Thanks in advance. Any comments are welcome!

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    $\begingroup$ You certainly want $\dim(X)>0$: if $\dim(X)=0$ then the projective limit has topological dimension $0$. $\endgroup$
    – YCor
    Commented Apr 14, 2021 at 13:05
  • $\begingroup$ @YCor Yes, I changed this error. Thanks for pointing it out. $\endgroup$
    – user119197
    Commented Apr 14, 2021 at 13:38

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