Open segment

Remarks

Example: on the real line, $(0, 1)$ in this sense is the open interval of numbers strictly between $0$ and $1$; the point $1/4$ is in it with $t = 1/4$.

Non-example: the endpoint $x$ of a line segment $[x, y]$ with $x \ne y$ is not in $(x, y)$, since $(1 - t) x + t y = x$ forces $t (y - x) = 0$, so $t = 0$, which is not allowed.

If $x \ne y$, then $(x, y) = [x, y] \setminus \set{x, y}$. If $x = y$, then $(x, y) = \set{x}$, so the open segment is not empty.