(We will be talking about the set of real numbers (0.1] hence all definitions are in terms of real numbers. A real number is an element of the complete ordered field.)

Start with a definition

There is a gap between 0 and a set of real numbers S iff there is a real number x, such that [0,x] does not contain an element of S.

Do you agree with this definition?

--
William Hughes

On Friday, May 28, 2021 at 5:41:53 AM UTC-7, William wrote:
> (We will be talking about the set of real numbers (0.1] hence all definitions are in terms of real numbers. A real number is an element of the complete ordered field.)
>
> Start with a definition
>
> There is a gap between 0 and a set of real numbers S iff there is a real number x, such that [0,x] does not contain an element of S.
>
> Do you agree with this definition?

That depends on how you treat intervals [a, b] where b < a

For instance, this reference says that those intervals are empty, in which case, by your criterion, there is no gap between 0 and the interval [-42, -17] :

https://en.wikipedia.org/wiki/Interval_(mathematics)#Classification_of_intervals

On 5/28/2021 9:42 AM, FredJeffries wrote:
> On Friday, May 28, 2021 at 5:41:53 AM UTC-7, William wrote:
>> (We will be talking about the set of real numbers (0.1] hence all definitions are in terms of real numbers. A real number is an element of the complete ordered field.)
>>
>> Start with a definition
>>
>> There is a gap between 0 and a set of real numbers S iff there is a real number x, such that [0,x] does not contain an element of S.
>>
>> Do you agree with this definition?
>
> That depends on how you treat intervals [a, b] where b < a
>
> For instance, this reference says that those intervals are empty, in which case, by your criterion, there is no gap between 0 and the interval [-42, -17] :
>
> https://en.wikipedia.org/wiki/Interval_(mathematics)#Classification_of_intervals
>

nope you miss read, that is for [b,a] and b < a

good link, worth refreshing on definitions

Contents

1 Terminology
1.1 Note on conflicting terminology
2 Notations for intervals
2.1 Including or excluding endpoints
2.2 Infinite endpoints
2.3 Integer intervals
3 Classification of intervals
4 Properties of intervals
5 Dyadic intervals
6 Generalizations
6.1 Multi-dimensional intervals
6.2 Complex intervals
7 Topological algebra
8 See also
9 References
10 Bibliography
11 External links

forgot about Dyadic intervals
wonder how WM does on connectedness, or continousness ?