How to prove:
When $F$ is a non-Archimedean local field.
then group $G = GL_n(F)$ is an open subset of $Mn(F)$?
The problem is from:
Let $n$ be an integer. The vector space $ F^n = F ×···× F $ carries the product topology, relative to which it is a locally profinite group. As a special case, the matrix ring $M_n(F)$ is a locally profinite group under addition, in which multiplication of matrices is continuous. The group $G = GL_n(F)$ is an open subset of $M_n(F)$; inversion of matrices is continuous, so G is a topological group.
Which said $G = GL_n(F)$ is an open subset of $M_n(F).$
$F$ is a non-Archimedean local field,and had proved $F $ as a additive group is a locally profinite group.
Thank you for sharing your idea!