RE: [P-1788]: Re objective == infinity
I am not proposing any new definitions, I am just explaining why the term "closed interval" can be ambiguous.
To avoid all these problems, we can simply add in the preamble that we consider closed intervals, i.e., intervals which are closed sets. The notion of a closed set is already unambiguous.
-----Original Message-----
From: stds-1788@xxxxxxxx [mailto:stds-1788@xxxxxxxx] On Behalf Of Vincent Lefevre
Sent: Sunday, September 30, 2012 6:37 PM
To: stds-1788
Subject: Re: [P-1788]: Re objective == infinity
On 2012-09-30 12:23:20 -0600, Kreinovich, Vladik wrote:
> I think the confusion is between
>
> * the notion of a closed set, which is well-defined and understood the
> same way in all math classes, and
>
> * the high-school notions of closed, open, and semi-closed intervals which in different textbooks may be interpreted differently for infinite intervals.
>
> If a textbook defines a closed interval as an interval that contains
> endpoints,
This is still ambiguous. What do you mean by "an interval that contains endpoints"? An interval that contains several endpoints (i.e. >= 2) or an interval that contains at least one endpoint (the plural is sometimes used when the number isn't known, but is >= 1).
Some textbooks would say that a closed interval contains its endpoints, which is again something else.
> then
>
> * [0, infinity) is not a closed interval, but
>
> * it is a closed set.
[0, infinity), as an interval of real numbers, has only one endpoint: 0.
Thus it contains its endpoints (which consist of the only endpoint 0).
Then, depending on the definition of a closed interval, you could see it as a closed interval or not.
Not everyone agrees on what a closed interval is (except for bounded intervals of real numbers, which match the topological definition of closed set). So, this notion should be avoided for unbounded intervals.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/> 100% accessible validated (X)HTML - Blog: <http://www.vinc17.net/blog/>
Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)