  We introduce a new lower-Vietoris-type hypertopology in a way similar to that
with which a new upper-Vietoris-type hypertopology was introduced in G. Dimov
and D. Vakarelov, "On Scott consequence systems", Fundamenta Informaticae, 33
(1998), 43-70. (it was called there {\em Tychonoff-type hypertopology}). We
study this new hypertopology and, in particular, we generalize many results
from E. Cuchillo-Ibanez, M. A. Moron and F. R. Ruiz del Portal, "Lower
semifinite topology in hyperspaces", Topology Proceedings, 17 (1992), 29-39. As
a corollary, we get that for every continuous map $f:X\longrightarrow X$, where
$X$ is a continuum, there exist a subcontinuum $K$ of $X$ such that $f(K)=K.$
