Between is the relation of a point 
to distinct collinear points
and 
for which 
. A number of Euclid's
proofs depend on the idea of betweenness without explicit mentioning it.
All points on a line segment excluding the endpoints lie between the endpoints.
Let 
be a partially
ordered set, and let 
. If 
,
then 
is said to be between 
and 
. If 
in 
and there is no 
that is between 
and 
, then 
covers 
. Conversely, if 
covers 
, then no 
is between 
and 
