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 ]
 Chas Brown Posts: 2,270 Registered: 12/6/04
Re: Another AC anomaly?
Posted: Dec 15, 2009 1:35 AM

On Dec 14, 5:28 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> Ilmari Karonen <usen...@vyznev.invalid> writes:
> > ["Followup-To:" header set to sci.math.]
> > On 2009-12-14, Jesse F. Hughes <je...@phiwumbda.org> wrote:

> >> "Jesse F. Hughes" <je...@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.
>

I'd say you were right, but...

Cheers - Chas

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