Re: [P-1788]: Re objective == infinity
On Sep 30 2012, Arnold Neumaier wrote:
Eh? When I did mathematics, a closed interval was one where any
countable set of elements within it had a unique lowest upper bound.
What form of closure are you using?
A closed subset of a metric space is one that contains with every point
a full ball centered at it. Equivalent is that it contains all its limit
points. This makes [0,Inf] closed.
Grrk. But this interval ISN'T [0,Inf] in the interval notation I learnt,
but [0,Inf), because it doesn't include the limit point of the sequence of
integers (for example). But at least I now understand which definition
you are using.
http://en.wikipedia.org/wiki/Closed_set explicitly says that [1,Inf] is
closed, though it uses the notation [1,Inf), which in our 1788
conventions denotes the same set.
It's closed under some countable operations and not others, which is
the problem, and is the reason that there are several different
definitions of closure used in this area.
All this is standard mathematical terminology.
(Wikipedia frequently has unconspicuous errors of this sort, and
different articles may contradict each other on fine points. Maybe
someone wants to correct the Wikipedia article on closed intervals.)
A far bigger problem is that too much of it is written by people who
don't realise that even standard mathematical terminology varies with
the field.
Regards,
Nick Maclaren.