Alex Fekken (5/19/2013)
This could go on for a long time... 🙂
yes, it could. :w00t:
The balls are no longer compact in infinite dimensional spaces (where many interesting applications, e.g. quantum mechanics, are) unless you use very special (weak) topologies.
we maybe don't agree on what topologies are "weak". (closed) Balls in an infinite-dimensional complete metric space are compact. Balls in an incomplete metric space are not (even in a simple 1-dimensional case like the rational line, balls are not compact). I haven't looked at the topologies used in QM but I suspect it will be lack of completeness rather than infinite dimension that makes balls incompact. Of course Borel's proof of compactness of totally bounded closed sets in Euclidean space breaks down with infinite dimensions, and I think that even adding completeness in leaves one needing something analogous to the Lowenheim-Skolem theorem or Tychonoff's theorem (that might be the best starting point) - or maybe a weak version of the axiom of choice would be enough (anyway I'm confused as to whether Tychonoff's theorem implies AC or not - I think it doesn't, but I'm not sure, and all my maths books are in the UK, four hours flight away).
I think (and hope) that I have made my point, which is not that my intuition is "better" than yours, just that you could argue either way, depending on context and/or background.
Oh, yes, quite definitely - it is a matter of how one reacts to "between" and there's nothing that would make everyone react the same way. My point really was that there's nothing inherently counter-intuitive in inclusion of bounds, and I didn't want to suggest that there would be something inherently counter-intuitive in their exclusion (although in the discrete case there would be for me, there might not be for something else).
Tom