I understand what an open ball is (and surely if you read my original post carefully this would be clear), I just don't think it actually makes the discussion c

I understand what an open ball is (and surely if you read my original post carefully this would be clear), I just don't think it actually makes the discussion clearer. In particular, when explaining something to people, starting with creating unfamiliar concepts is a Bad Idea if those concepts don't have a significant payoff. In particular, because this is not a discussion about arbitrary spaces, the use of the word "Ball" is counter-intuitive. (But I admit I am probably biased by my own experience.)