  We introduce different Finsler metrics on the space of smooth K\"ahler
potentials that will induce a natural geometry on various finite energy classes
$\mathcal E_{\tilde \chi}(X,\omega)$. Motivated by questions raised by R.
Berman, V. Guedj and Y. Rubinstein, we characterize the underlying topology of
these spaces in terms of convergence in energy and give applications of our
results to existence of K\"ahler-Einstein metrics on Fano manifolds.
