Is the convex hull of union of a line and a point an open set?

Question

Given a line $l$ and a point $A$ not on the line, is the convex hull of the two an open set? As per my understanding, the hull will be all points between lines $l$ and line $m$ which passes through $A$ and is parallel to $l$, including line $l$ but not line $m$, except the point $A$. Is this a closed set?

Answer 1

No, in the plane, take say the line $y=0$ and the point $(0,1)$, then the point $(1,1)$ would on your line $m$. $(1,1)$ can be written as the limit of points in the convex hull but does not belong itself to the convex hull, i.e.,

$$(1,1) = \lim_n (1-\frac{1}{n}) (0,1) + \frac{1}{n} (n,0),$$

where $(1-\frac{1}{n}) (0,1) + \frac{1}{n} (n,0)$ belongs to the convex hull.