$K$ a local field, the spaces $K^n : n = 1, 2, ...$ are not homeomorphic

Question

Let $K$ be a local field. If $K$ is archimedean, then one can distinguish the spaces topological spaces $K, K^2, K^3$ etc. by their homotopy groups. How can one distinguish these spaces when $K$ is nonarchimedean?

Answer 1

As someone pointed out to me today, these spaces are actually homeomorphic when $K$ is nonarchimedean and of characteristic zero!

For example, take $K = \mathbb{Q}_p$. It is known that $\mathbb{Z}_p$ is homeomorphic to the cantor set $C$, and $C \simeq C^2$. Thus there is a homeomorphism $\phi: \mathbb{Z}_p \rightarrow \mathbb{Z}_p \times \mathbb{Z}_p$.

Let $k \leq 0$ be an integer. Define a homeomorphism $\phi_k: p^k\mathbb{Z}_p \rightarrow p^k\mathbb{Z}_p \times p^k\mathbb{Z}_p$ by $x \mapsto p^k \phi(\frac{1}{p^k}x)$.

Now $\mathbb{Q}_p$ is the direct limit of the directed system of topological spaces $\mathbb{Z}_p \subseteq p^{-1}\mathbb{Z}_p \subseteq \cdots$, and similarly $\mathbb{Q}_p \times \mathbb{Q}_p$ is the direct limit of the system $\mathbb{Z}_p \times \mathbb{Z}_p \subseteq p^{-1}\mathbb{Z}_p \times p^{-1}\mathbb{Z}_p \subseteq \cdots$.

Then $\phi = \varinjlim \phi_k$ extends to a homeomorphism $\mathbb{Q}_p \rightarrow \mathbb{Q}_p^2$.