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On 09/30/2012 01:25 PM, N.M. Maclaren wrote:
On Sep 30 2012, Vincent Lefevre wrote:> Right. So there are no closed unbounded intervals. No, [1,+inf] is closed, because it is the complement of the open set [-inf,1[. But it is not a compact.Note: in case this is not clear, I'm taking the topology on R (not Rbar), because we are talking about intervals of real numbers and Entire is R (not Rbar).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.
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.
Wikipedia uses the more elementary description that a closed interval includes its endpoints.
This is correct only for bounded intervals. (The Wikipedia articlehttp://en.wikipedia.org/wiki/Closed_interval partially assumes boundedness without saying so explicitly. This can also be seen from its definition of the midpoint, which makes sense only in the bounded case.)
An unbounded interval is closed iff it contains its finite endpoints. 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.)
Arnold Neumaier