We study the frame bundle $F(\ell)$ of a vector bundle $\ell$ with

fibre type a projective finitely generated module over a (topological)

algebra ${\mathbb A}$. The topological-algebraic structure of

${\mathbb A}$ is crucial in our considerations. In fact, if

${\mathbb A}$ is a $Q$-algebra, $F(\ell)$ is a smooth principal bundle

and its connections correspond bijectively to

${\mathbb A}$-connections on $\ell$, as in the case of Banach bundles.

If ${\mathbb A}$ is not a $Q$-algebra, then $F(\ell)$ is only a

topological principal bundle. However, it can be provided with

sheaf-theoretic entities, legitimately called connections, which

essentially describe the connections of $\ell$. As a result, the

geometry of our bundles can be reduced to a topological-algebraic

context embodying all the previous cases and giving an example of the

effectiveness of the methods of the ``abstract differential geometry''

initiated in \cite{MAL;VS} for ``vector sheaves'' and further applied

and extended for ``principal sheaves'' in \cite{VASS;Conn}, here

combined with topological algebra theory.