Line segment

Remarks

Example: in $\R^2$, the point $(1, 2)$ lies in $[(0, 0), (2, 4)]$, since $(1, 2) = \tfrac12 (0, 0) + \tfrac12 (2, 4)$ with $t = 1/2 \in [0, 1]$.

Non-example: the point $(3, 6)$ lies on the line through $(0, 0)$ and $(2, 4)$, but $(3, 6) = (1 - t)(0,0) + t (2, 4)$ forces $t = 3/2$, which is not in $[0, 1]$. So $(3, 6) \notin [(0,0), (2,4)]$.

The segment $[x, y]$ is the set of convex combinations of $x$ and $y$, and $[x, y] = [y, x]$. If $x = y$, then $[x, y] = \set{x}$.