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Topology: Difference between Dedekin-cuts and cantor-bendixson in countability
Posted:
Sep 9, 2006 10:48 PM
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It seems my question was to complicated for this group, so I will
re-formulate it into a series of questions:
1) How do you prove that the set C_A:=(A - A') is countable for all
sets A subset of Real numbers, where A' is the cantor-bendixson-derivative
A':={x in A | x in closure_R(A - {x})}?
2) How do you prove that the set of lower-half dedekin-cuts
D:={A subset Q|for-all x in A,y in Q: y<x => y in A} is uncountable?
1a) How do you prove that for all A subset of real numbers, there
exists a function f:C_A -> D which is injective, and for-all
x in C_A: x in closure_R(f(x)) and there exists q in f(x) with q>x,
and for-all y in C_A: x<y => there exists q in f(y) with q<y and
q not in f(x)?
1b) How do you solve 1) by proving that S_A:={f(x)|x in C_A} which is
subset of D is really countable, provided you have solved 1a)?
2a) how do you prove that there exists a function g:D->R which is
surjective, and for-all d in D: g(d) is border-point of closure_R(d)?
2b) Without using anything similar to the function g defined in 2a),
how do you then prove 2)?
3) how do you prove that for all uncountable sets C subset of R,
there exists no A subset of R, such that
C==C_A(=={x in A | x not-in closure_R(A - {x})})?
3a) provided you have solved 1), how do you solve 3) without
re-using parts of the solution for 1)? (i.e. without the "law"
that everytime an assumption is made and one could prove that
it leads to a contradiction, the negation of this assumption
must be true!)
4) how do you prove that for-all O set of simply-connected subsets of Q:
either O is countable or there exists x,y in O such that intersection of
x and y is non-empty (but not both)?
4a) O:={x | there exists y in D: x subset of y, there exists no z in D:
z true subset of y and x has non-empty intersection with z} is countable?
consider h:D->O with h(d):={x subset of d | there exists no y in D:
y true subset of d and x has non-empty intersection with y} for-all d in D,
where "x true subset of y" is defined as x subset of with x!=y. h is
surjective. for all d in D: h(d) == empty-set iff for-all x subset of d
there-exists y in D: y true subset of d and x has non-empty intersection
with y. How do you prove that there exists h(d)==empty-set when assuming
that above x is a singelton -- in ZFC?
Of course R is the set of Real Numbers, Q is the set of rational numbers,
closure_R is the usual closure in the topology of R, in all other cases
the topology I'm working on is the usual topology on Q.
--
Better send the eMails to netscape.net, as to
evade useless burthening of my provider's /dev/null...
P
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