any help will be appreciated for this problem. I have an idea what the solution should be, but I'm unsure if I have enough justification.
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Let p = (0,1) and q = (0,-1). Let X be the subset of that is formed by intersecting all the closed Euclidean balls that contain p and q. What is X?
Solution (so far):
We can immediately exclude closed Euclidean balls that do not have p and q as a boundary point, since there exist Euclidean balls contained in them, and that if X is contained in Y, the intersection is X.
With some basic geometry, I've shown that there are three balls that have p and q as boundary points: the unit ball centered at the origin, and balls of radius centered at (-1,0) and (1,0).
Now, I want to say that the intersection between the last two balls mentioned is X, but I cannot convince myself that is the case. Am I doing something wrong?
That is what was said: " intersecting all the closed Euclidean balls that contain p and q."
If it had said " intersecting all the closed Euclidean balls that contain p or q", then I would have interpreted it the way you did. In this case, the intersection is the Euclidean line segment having p and q as endpoints.