# Topological dimension of $p$-adic manifolds

What is the topological dimension of a (locally analytic) $p$-adic manifold over a non Archimedean field $K$?

Is the topological dimension of $K^n$, $n$?

• Sep 2, 2018 at 15:38
• If you want topological dimension to be the "expected dimension", so the dimension of the $n$-dimensional affine space to be $n$, you can consider the Berkovich analytic space, which has much more points that $K^n$. This is one of the reasons these spaces were introduced. Sep 3, 2018 at 10:22
• Just for the record, at least for discretely valued $K$, the spaces $K$ and $K^n$ are homeomorphic for any $n$. Sep 3, 2018 at 16:30

$p$-adic numbers are locally compact, Hausdorff and totally disconnected (see this nLab page), hence they are zero-dimensional. This means that---at least naively---topological dimension of $p$-adic manifolds doesn't work as you'd expect from real or complex manifolds. However, there are ways to do analytic geometry over the $p$-adic numbers, see e.g. this stackexchange question and the references on this nLab page.