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On 05.11.2024 18:25, Jim Burns wrote:On 11/4/2024 12:32 PM, WM wrote:
⎛ When the intervals are end.to.end.to.end,>The intervals together cover a length of less than 3.>
The whole length is infinite.
Therefore there is plenty of space for
a point not in contact with any interval.
⎛ Assuming the covering intervals are translated
⎜ to where they are end.to.end.to.end,
⎜ there is plenty of space for
⎝ not.in.contact exterior points.
This plentiness does not change
when the intervals are translated.
'Exterior' seems like a good way to sayI mean 'exterior' in the topological sense.>
>
For a point x in the boundary ∂A of set A
each open set Oₓ which holds x
holds points in A and points not.in A
The intervals are closed with irrational endpoints.
Are any of these points.outsideEach of {...,-3,-2,-1,0,1,2,3,...} is>
the midpoint of an interval.
There can't be any exterior point
a distance 1 from any interval.
>
There can't be any exterior point
a distance ⅟2 from any interval.
Nor ⅟3. Nor ⅟4. Nor any positive distance.
Nice try.
But there are points outside of intervals,
and they are closer to interval endsShouldn't I be pointing that out
than to the interior, independent of
the configuration of the intervals.
Note thatYes, less than 2³ᐟ²⋅ε
only 3/oo of the points are inside.
Positive ℕ⁺ holds countable.to from.1An exterior point which is not>
a positive distance from any interval
is not an exterior point.
Positive is what you can define,
but there is much more in smaller distance.Distances are positive or zero.
Remember the infinitely many unit fractionsFor each of the infinitely.many unit fractions
within every eps > 0 that you can define.
Did you intend to write "interior"?Therefore,>
in what is _almost_ your conclusion,
there are no exterior points.
There are 3/oo of all points exterior.
We are only toldInstead, there are boundary points.>
For each x not.in the intervals,
each open set Oₓ which holds x
holds points in the intervals and
points not.in the intervals.
x is a boundary point.
The intervals are closed
Explain why.>The rationals are dense>
Yes.
Each multi.point interval [x,x′] holds
rationals.
>but the intervals are not.>
No.
Each multi.point interval [x,x′] holds
ε.cover intervals.
Therefore not all rationals are enumerated.
It contradicts a non.empty exterior.proves that>
the rationals are countable.
Contradiction.
Something of your theory is inconsistent.Your intuition is disturbed by
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