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On 11/6/2024 11:24 AM, WM wrote: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.On 06.11.2024 15:22, Jim Burns wrote:On 11/6/2024 5:35 AM, WM wrote:Yes.>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
and is not in contact.You don't know that.
The _union_ ofWe use only intervals, not limits. A point is in an interval or it is not.
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.
"My" boundary is a definition.But it is irrelevant here like the offside rule in soccer.
The term "boundary" helps clarifyNo,it is not confusing at all. For every interval we can decide whether a poin is inside or outside.
what I admit is a confusing situation.
https://en.wikipedia.org/wiki/Boundary_(topology)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.
⎛ 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 :
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