Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 Jesse F. Hughes Posts: 9,776 Registered: 12/6/04
Re: Another AC anomaly?
Posted: Dec 14, 2009 8:28 PM

Ilmari Karonen <usenet2@vyznev.invalid> writes:

> ["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.

Oh. Okay, but if I'm right, it was only coincidence. So, perhaps I
was butt-right.

--
Jesse F. Hughes
"We need to counter the shockwave of the evildoer by having individual
rate cuts accelerated and by thinking about tax rebates."
-- George W. Bush, Oct. 4, 2001

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