Consider $E$ a compact metric space and the space of continuous paths from $[0,T]$ to $E$ and call it $\Omega$ and let $\pi_t$ be the projections from $\Omega$ to $E$: $\pi_t(w) = w(t)$.
Is the topology making all the maps $\pi_t$ continuous the topology of pointwise convergence?
In contrast this should be different to the topology of uniform convergence on compact subsets, but are the $\sigma$-algebras associated with each of the previous topologies equal?
Am I right in consider the first topology as a weak topology on the Banach space $\Omega$ when endowed with the sup norm? Is the second then just the strong topology, and then, the $\sigma$-algebra generated by them should be the same as is usual in this setting?
Bye.