Re: [P-1788]: Re objective == infinity
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?
Wikipedia uses the more elementary description that a closed interval
includes its endpoints. If we are talking about the unexpected
reals, then what I said is correct and [1,+inf] is NOT a closed
interval, because +inf is not an element in the set of values defined
by the interval.
It does mean that certain functions may need to give an error if
passed an infinity as an argument, but that's not a major problem.
Not sure what you mean here. Functions take intervals as inputs
(except constructors).
Sooner or later, someone will want the predicate to enquire if a
value is an element of the set. If infinity is not a valid value,
then that enquiry is erroneous.
Regards,
Nick Maclaren.