  In trying to generalize the classic Sylvester-Gallai theorem and De
Bruijn-Erd\H{o}s theorem in plane geometry, lines and closure lines were
previously defined for metric spaces and hypergraphs. Both definitions do not
obey the geometric intuition in the sense that two lines (closure lines) may
intersect at more than one point, and one line (closure line) might be the
proper subset of another. In this work, we study the systems where one or both
of the configurations are forbidden. We note that when any two lines intersect
in at most one point, the two classic theorems extend in any metric space. We
study the metric spaces induced by simple graphs where no line is a proper
subset of another, and show that the least number of lines for such a graph
with $n$ vertices is between the order of $n^{4/3}$ and $n^{4/3} \ln^{2/3} n$.
