> A zero-length line segment is a point.
The Wikipedia definition of a line segment says that it is bound by two endpoints.[1] It provides a reference to Planet Math that goes into specifics.[2] In this page, it is made clear that the two endpoints cannot be equal. Planet Math provides an equation for a closed[3] line segment:
L = {a + tb | t in [0, 1]}
This means that a line segment can be expressed as all of the points
a + tb, where
t is the range [0, 1] (which contains 0, 1 and all the points between them). It also limits
a, b as real or complex numbers with
b != 0. In other words, any line segment is just "morphing" the basic 0 to 1 range, with
a shifting it and
b scaling it.
Now, since b can't be 0, and 0 != 1, you're not going to get the endpoints of the range 0 and 1 to equal each other no matter how you scale them with b or shift them with a.[4] In other words, line segments will always have length.
Because you cannot get the endpoints to equal each other, a point cannot be thought of as a line segment under what appears to be the common definition.
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[1] https://en.wikipedia.org/wiki/Line_segment
[2] http://planetmath.org/encyclopedia/LineSegment.html
[3] Open is the exact same thing, except with (0, 1) instead of [0, 1] and it doesn't include the points 0 and 1.
[4] Of course, if b = 0 was allowed, you could just say that points are where b = 0 for all a.