Re: Matrices and topologies

Re: Matrices and topologies

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"

(a_{1},b_{1}) x.....(a_{n},b_{n}) (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 k^{n} generated by:

U_{1}x...xU_{n}, where each U_{j} is open in k (in fact, the U_{j} 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.