Re: Boundary of a Union of Sets

In article <1147198784.407562.298120@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<mikharakiri_nospaum@xxxxxxxxx> wrote:

Arturo Magidin wrote:
In article <1147195710.855484.66900@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
MimsyBoro <eytan@xxxxxxxxxxxxxxx> wrote:
I do mean that a bounday is "points in the
closure that are not in the interior" and I didn't ask if they were
equal but is the boundary of a union a *subset* of the union of the
boundaries.

Oh, quite right.

By the way:

http://groups.google.com/support/bin/answer.py?answer=14213&topic=250

So you are asking if closure(Union A_i) - interior(Union A_i) is
contained in Union (Closure(A_i) - Interior(A_i)).

No. Let f:N->Q be any enumeration of the rationals. Let A_i =
{f(i)}. Then closure(A_i) = A_i, Interior(A_i) = {}, so the union of
the boundary points is just Q.

However, the closure of the union of the A_i is all of R, and interior
of the union of the A_i is empty, so the boundary points of the union
are all of R, which is not a subset of Q.

The question is about union of 2 sets, not countably many.

Looks like misreading is contagious.

The original poster said it was clear that the boundary of the union
of two sets is contained in the union of the boundaries of the two
sets. I misread her assertion as equality, which is what she
corrected.

She then asked, in the original post, whether the result was also true
for uncountable unions. That is, whether the boundary of an
(uncountable) union of sets is contained in the (uncountable) union of
the boundaries of the sets. In fact, the result already fails for
countable unions, as the example shows; you can get an uncountable
union by adjoining however many copies of the empty set you want. Or
you can take that example and modify it to obtain an example with
a pairwise disjoint nonempty uncountable family of subsets of R for
which the boundary of the union is not contained in the union of the
boundaries.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.

Relevant Pages

  • Re: Boundary of a Union of Sets
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  • Re: Boundary of a Union of Sets
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  • Re: Boundary of a Union of Sets
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