Let Mat(n,k) be the set of nxn matrices with entries in the topological field k. This set can be identified with k^n^2, so it gets a topology.
Let GL(n,k) be the subspace of nxn invertible matrices
My question is, how are open sets defined in this topological space, Mat(n,k)? And is the subspace GL(n,k) defined similarly, the intersection of GL(n,k) with all open sets of Mat(n,k)?
for a finite-dimensional vector space over R, of dimension n, we can use the "box" topology which is generated by the base of open "boxes"
(a1,b1) x.....(an,bn) (it's easiest to visualize this when n = 3).
for an arbitrary topological field k the open sets of k are not necessarily so easily described, but we still have a topology on kn generated by:
U1x...xUn, where each Uj is open in k (in fact, the Uj can be basis element of the topology of k).
and yes, this allows a topology to be defined on GL(n,k) via the relative topology.