Proving SL_2(C) Homeomorphic to SU(2)xT & Simple Connectedness

Using the QR decomposition (the complex version) I want to prove that ##SL_2(C)## is homeomorphic to the product ##SU(2) × T## where ##T## is the set of upper-triangular 2×2-complex matrices with real positive entries at the diagonal. Deduce that ##SL(2, C)## is simply-connect.

So, I can define a map ##SU(2)×T–>SL(2,C)## given by: ##(u,t) –> ut## where ##u \in SU(2)## and ##t \in T## (from QR decomposition we have that each ##A## in ##GL(2,C)## can be written as ##ut## where ##u \in U(n)## and ##t \in T## (##T## mentioned above)).
We have that the intersection ##SU(2)\cap T=1##.
This map is injective:
if ##u_1t_1=u_2t_2## then this gives
##u_2^{-1}u_1=t_2t_1^{-1} \in SU(2)\cap T## so ##u_1=u_2## and ##t_1=t_2##.
Can I say that this map is surjective by the QR decomposition?
why is this map and its inverse contiuous?

Thanks for help!

Where does the determinant of ##t## go to in case it is not ##(a,a^{-1})?##

Maybe one of these mappings here serves you better:
https://www.physicsforums.com/insig...c-part/#3-The-3-Sphere-or-Glome-or-SU2mathbbC