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On 09/30/2012 02:41 PM, N.M. Maclaren wrote:
On Sep 30 2012, Arnold Neumaier wrote: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.
By the P1788 definition, [0,Inf] is the set of all real numbers between 0 and Inf, which excludes Inf as it is no real number.
Thus [0,Inf]=[0,Inf). The integers have no real limit point. Inf is a limit point only in the extended reals.
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.
Closed interval is defined (and universally understood) with closed in the topological sense.
A closed interval is not closed under many things, for example not under squaring or division by two. ''closed under'' is therefore a concept quite different from ''closed''.
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.
The conventions about closed intervals are very standard, and don't vary with the field. This stuff is taught to all math students in the first semester.