## The connection as a choice of horizontal subspace?

Hi,
I'm trying to understand the fiber bundle formulation of gauge theory at the moment, and I'm stuck on the connection. Every reference I've found introduces the idea of a connection on a principle bundle as a kind of partitioning of the tangent space at all points in the total space into a "vertical space" and a "horizontal space". The vertical space Vp consists of vectors in TpP which are also tangent to the fiber at p, and the horizontal space Hp is a set of vectors such that Vp+Hp=TpP.
What I don't understand is why finding Vp doesn't uniquely specify Hp. It should be possible to construct TpP without defining a connection, right? If so, wouldn't Hp just be every element of TpP that is not also in Vp? I don't see how we are free to make this partition ourselves. Where am I going wrong?