  We give a general version of Bryc's theorem valid on any topological space
and with any algebra $\mathcal{A}$ of real-valued continuous functions
separating the points, or any well-separating class. In absence of exponential
tightness, and when the underlying space is locally compact regular and
$\mathcal{A}$ constituted by functions vanishing at infinity, we give a
sufficient condition on the functional $\Lambda(\cdot)_{\mid \mathcal{A}}$ to
get large deviations with not necessarily tight rate function. We obtain the
general variational form of any rate function on a completely regular space;
when either exponential tightness holds or the space is locally compact
Hausdorff, we get it in terms of any algebra as above. Prohorov-type theorems
are generalized to any space, and when it is locally compact regular the
exponential tightness can be replaced by a (strictly weaker) condition on
$\Lambda(\cdot)_{\mid \mathcal{A}}$.
