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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
Date: Wed, 6 Nov 2024 20:50:23 +0100
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On 06.11.2024 20:08, Jim Burns wrote:
> On 11/6/2024 11:24 AM, WM wrote:
>> On 06.11.2024 15:22, Jim Burns wrote:
>>> On 11/6/2024 5:35 AM, WM wrote:
> 
>>>> The intervals are closed with irrational endpoints.
>>>
>>> 'Exterior' seems like a good way to say
>>> 'not in contact'.
>>
>> Every point outside is not an endpoint
> 
> Yes.
> 
>> and is not in contact.
> 
> You don't know that.

 From every positive point we know that it is not 0 and not in contact 
with (-oo, 0]. Same for every point not in an interval.
> 
> The _union_ of
> arbitrarily.many _open_ sets
> is an open set.
> However,
> the _union_ of
> these infinitely.many _closed_ intervals
>   with irrational endpoints
> is an open interval
>   with rational endpoints.

We use only intervals, not limits. A point is in an interval or it is not.

> "My" boundary is a definition.

But it is irrelevant here like the offside rule in soccer.

> The term "boundary" helps clarify
> what I admit is a confusing situation.

No,it is not confusing at all. For every interval we can decide whether 
a poin is inside or outside.
> https://en.wikipedia.org/wiki/Boundary_(topology)
> ⎛ It is the set of points p ∈ X such that
> ⎜ every neighborhood of p contains
> ⎜ at least one point of S and
> ⎝ at least one point not of S :

But there are no points outside of intervals because, if Cantor has 
enumerated all rationals, then all rationals are caught and irrationals 
cannot be outside. Therefore a boundary is excluded. Points p can only 
exist insideof intervals.

Further: For every point x and for every interval we can find x - e 
where e is the endpoint of an interval. No boundary is useful, in 
particular because the infinite positive real axis except 3 unit 
intervals cannot be covered by the blurred boundary.

Regards, WM
> 
>