> However, 'ball' is from geometry, while 'neighborhood' is from analysis. They overlap, but they're not to be confused, because 'neighborhood' is used to empha

> However, 'ball' is from geometry, while 'neighborhood' is from analysis. They overlap, but they're not to be confused, because 'neighborhood' is used to emphasize that one should not visualize it. You’re right, but in fairness I’ve seen multiple analysis textbook authors use terms like “circle” and “ball” to describe a neighborhood in topology. This is usually in the fashion, “...(in other words, E is a ball.” This is precisely the sort of thing I dislike, because aside from being technically incorrect like you say, it can (for that section) encourage a student to think, “Oh, why didn’t he just say that?” and skip the heavier definition -> theorem -> proof sequence preceding it, to the detriment of understanding later material.