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.
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Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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