Re: [P-1788]: Re objective == infinity
Nick,
Please see my inserted comments.
Baker
On 09/30/2012 06:25 AM, 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?
Topologically, closed means it contains all of its limit points, where
a limit point x of a set S is such that every open set in
the topology containing x contains a point of S other than x.
If we are looking at the topology on the reals (and not the
extended reals), infinity is not in the space, so it can't
be a limit point, and every other point in the space is finite,
so every open interval about it contains a point in the space.
Hence [1,+inf) is closed in the topology on the reals. (Note
that I prefer the notation [1,inf) to [1,inf] to emphasize that
inf is not actually an element, even though the interval is
closed.)
Actually, I see the practical importance to us of considering the
reals, rather than the extended reals, as determining how we
define things like [0.0] [1,inf). (Siegfried Rump pointed this
out to me on a bus in Okinawa several years ago.)
Wikipedia uses the more elementary description that a closed interval
includes its endpoints.
Not exactly. I quote:
"Closed set
From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article is about the complement of an open set.
For a set closed under an operation, see closure (mathematics). For other uses, see Closed (disambiguation).
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
[1][2] In a topological space, a closed set can be defined as a set which contains all its limit points.
In a complete metric space, a closed set is a set which is closed under the limit operation.
If we are talking about the unexpected
reals,
What are "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.
Huh?
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.
I hope not. We're trying to make things a smooth as possible.
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.
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