  The motivation of this paper is a suggestion by H\"ole of comparing the
notions of $\D$-boundedness and boundedness in Probabilistic Normed spaces
(briefly PN spaces), with non necessarily continuous triangle functions. Such
spaces are here called ``pre-PN spaces''. Some results on \v{S}erstnev spaces
due to B. Lafuerza, J. A. Rodriguez, and C. Sempi, are here extended to
generalized \v{S}erstnev spaces (these are pre-PN spaces satisfying a more
general \v{S}erstnev condition).
  We also prove some facts on PN spaces (with continuous triangle functions).
First, a connection between fuzzy normed spaces defined by Felbin and certain
\v{S}erstnev PN spaces is established. We further observe that topological
vector PN spaces are $F$-normable and paranormable, and also that locally
convex topological vector PN spaces are bornological. This last fact allows to
describe continuous linear operators between certain generalized \v{S}erstnev
spaces in terms of bounded subsets.
