In this example the family of curves appear to be continuous only with respect to the Hausdorff distance, since one of the curves is actually closed. In the problem I had in mind, the family of curves are continuous with respect to the C^1 norm as maps from an interval to R^n. So I do not see that the example you mention works in the sense that I intended. I will make the notion of continuity more explicit in the question.

I don't think that this example works because one could always include the point $B_t$ (I am not assuming that the number of points is minimal).