Let me try to help one more time. The number of points between any two unequal points (that is different points, if you have two points at one location you have in fact only one point) is infinite. There is no such thing as "adjoining points) since geometric points have no spatial dimensions. They have no length depth or width, they have only location. At any rate here is a quote from an online source:
"Let's assume that your end points are A(0,0) and B(1,0).
Try to find a point in between them. Ok you found C(0.4,0).
Now try finding another point. Will you ever NOT be able to find another point? No, because there are infinite numbers between 0 and 1 (just think of decimals).
In the same way there are infinite points in a line segment."
Why/How are there infinite points in a line segment?
If you have two points then by definition there is space between them and that is filled with an infinite number of points.