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Topic: Another AC anomaly?
Replies: 43   Last Post: Dec 21, 2009 8:08 AM

 Messages: [ Previous | Next ]
 Ilmari Karonen Posts: 311 Registered: 2/1/07
Re: Another AC anomaly?
Posted: Dec 14, 2009 4:27 PM

["Followup-To:" header set to sci.math.]
On 2009-12-14, Jesse F. Hughes <jesse@phiwumbda.org> wrote:
> "Jesse F. Hughes" <jesse@phiwumbda.org> writes:
>>
>> But the standard topology on N is the discrete topology, too! Thus,
>> the standard definition of sequence convergence on N is inherited via
>> the subspace topology from Set. That is, a sequence
>> {a_n | n in N} c N converges (in N) to m iff
>>
>> (E k)(A j > k) a_j = m.
>>
>> This is (unless I'm just butt-wrong) the same as the definition of
>> sequence convergence on Set restricted to the subspace N.

>
> Yeah, well, I am just butt-wrong, ain't I?

Well, not really. That's not the same as the definition of general
set convergence, but I do believe the two definitions are equivalent
for sequences of natural numbers, at least under any of the usual
set-theoretic constructions of the naturals.

In particular, under the standard construction of the naturals, where
0 = {} and n+1 = n union {n}, I believe the two definitions of lim sup
and lim inf also match: this is due to the fact that, for the natural
numbers m and n under this construction, m is a subset of n if and
only if m <= n.

--
Ilmari Karonen
To reply by e-mail, please replace ".invalid" with ".net" in address.

Date Subject Author
12/12/09 Jesse F. Hughes
12/13/09 K_h
12/14/09 Dik T. Winter
12/14/09 K_h
12/15/09 Dik T. Winter
12/15/09 K_h
12/16/09 Dik T. Winter
12/16/09 K_h
12/17/09 Dik T. Winter
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/21/09 Dik T. Winter
12/14/09 Dik T. Winter
12/14/09 K_h
12/15/09 Dik T. Winter
12/15/09 Dik T. Winter
12/15/09 K_h
12/16/09 Dik T. Winter
12/17/09 K_h
12/17/09 Dik T. Winter
12/18/09 K_h
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/18/09 Dik T. Winter
12/18/09 K_h
12/15/09 K_h
12/16/09 Jesse F. Hughes
12/17/09 Dik T. Winter
12/17/09 Jesse F. Hughes
12/16/09 Dik T. Winter
12/15/09 ross.finlayson@gmail.com
12/13/09 K_h
12/13/09 Jesse F. Hughes
12/13/09 Jesse F. Hughes
12/14/09 Ilmari Karonen
12/14/09 Jesse F. Hughes
12/15/09 Chas Brown