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Topic: Stone Cech
Replies: 49   Last Post: Mar 28, 2013 12:15 PM

 Messages: [ Previous | Next ]
 Butch Malahide Posts: 894 Registered: 6/29/05
Re: Stone Cech
Posted: Mar 22, 2013 12:08 AM

On Mar 21, 2:03 pm, quasi <qu...@null.set> wrote:
> Butch Malahide wrote:
> >quasi wrote:
> >> quasi wrote:
> >> >quasi wrote:
> >> >>quasi wrote:
> >> >>>Butch Malahide wrote:
> >> >>>>David C. Ullrich wrote:
> >> >>>>>Butch Malahide wrote
> >> >>>>> >William Elliot wrote:
> >> >>>>> >>David Hartley wrote:
> >> >>>>> >> >William Elliot wrote:
>
> >> >>>>> >> >>Perhaps you could illustrate with the five
> >> >>>>> >> >>different one to four point point
> >> >>>>> >> >>compactifications of two open end line
> >> >>>>> >> >>segements.

>
> >> >>>>> >> >(There are seven.)
>
> >> >>>>> >>Ok, seven non-homeomophic finite Hausdorff
> >> >>>>> >>compactications.

>
> >> >>>>> >How many will there be if you start with n segments
> >> >>>>> >instead of 2?

>
> >> >>>>> Surely there's no simple formula for that?
>
> >> >>>>> ...
>
> >> >>>> ...
>
> >> >>>>I wasn't necessarily expecting a *complete* answer, such
> >> >>>>as an explicit generating function. Maybe someone could
> >> >>>>give a partial answer, such as an asymptotic formula, or
> >> >>>>nontrivial upper and lower bounds, or a reference to a
> >> >>>>table of small values, or the ID number in the
> >> >>>>Encyclopedia of Integer Sequences, or just the value for
> >> >>>>n = 3. (I got 21 from a hurried hand count.)

>
> >> >>>For n = 3, my hand count yields 19 distinct
> >> >>>compactifications, up to homeomorphism.

>
> >> >>>Perhaps I missed some cases.
>
> >> >>I found 1 more case.
>
> >> >>My count is now 20.
>
> >> >I found still 1 more case.
>
> >> >So 21 it is!
>
> >> >But after that, there are no more -- I'm certain.
>
> >> Oops -- the last one I found was bogus.
>
> >> So my count is back to 20.
>
> >Hmm. I counted them again, and I still get 21.
>
> >4 3-component spaces: OOO, OO|, O||, |||.
>
> >7 2-component spaces: OO, O|, O6, O8, ||, |6, |8.
>
> >10 connected spaces: O, |, 6, 8, Y, theta, dumbbell, and the
> >spaces btained by taking a Y and gluing one, two, or all
> >three of the endpoints to the central node.

>
> Thanks.
>
> It appears I missed the plain "Y", but other than that,
> everything matches.
>
> So yes, 21 distinct types.

For n = 4 I get 56 types. If I counted right (very iffy), it may or
may not (probably not) be interesting to note that 2, 7, 21, 56 is the
beginning of OEIS sequence A202027. Probably just a coincidence.

"It is interesting to note that most statements beginning 'it is
interesting to note' are not interesting to note."

Date Subject Author
3/14/13 William Elliot
3/14/13 fom
3/15/13 fom
3/16/13 William Elliot
3/15/13 David C. Ullrich
3/17/13 William Elliot
3/17/13 David C. Ullrich
3/17/13 fom
3/18/13 David C. Ullrich
3/18/13 fom
3/18/13 David Hartley
3/19/13 William Elliot
3/19/13 David Hartley
3/19/13 William Elliot
3/20/13 Butch Malahide
3/20/13 David C. Ullrich
3/20/13 Butch Malahide
3/20/13 Butch Malahide
3/21/13 quasi
3/21/13 quasi
3/21/13 quasi
3/21/13 quasi
3/21/13 Butch Malahide
3/21/13 quasi
3/22/13 Butch Malahide
3/22/13 Butch Malahide
3/22/13 Butch Malahide
3/22/13 quasi
3/22/13 David C. Ullrich
3/22/13 David C. Ullrich
3/22/13 Butch Malahide
3/23/13 Butch Malahide
3/23/13 David C. Ullrich
3/23/13 David C. Ullrich
3/23/13 Frederick Williams
3/23/13 David C. Ullrich
3/23/13 Frederick Williams
3/22/13 Butch Malahide
3/23/13 David C. Ullrich
3/22/13 Butch Malahide
3/23/13 quasi
3/23/13 Butch Malahide
3/23/13 Butch Malahide
3/24/13 quasi
3/24/13 Frederick Williams
3/24/13 quasi
3/25/13 Frederick Williams
3/28/13 Frederick Williams
3/25/13 quasi
3/19/13 David C. Ullrich