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

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Stone Cech
Posted: Mar 22, 2013 10:25 PM
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On Mar 22, 4:34 am, quasi <qu...@null.set> wrote:
> Butch Malahide wrote:
> >Butch Malahide wrote:
> >>Butch Malahide wrote:
> >>>quasi 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 obtained 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),
>
> >> Found two more. Never mind!
>
> >And now I get 61. The hell with it.
>
> Ullrich predicted it (hopeless squared).
>
> For small n, say n < 10, it might be feasible to get the counts
> via a computer program, but my sense is that the development of
> such a program would be fairly challenging. If I get a chance,
> I may give it a try.


Now I get 2, 7, 21, 61, 180 for the first five values. Doesn't match
anything in the OEIS. I did the work slowly and carefully (took about
an hour and a half), and so I'm reasonably sure my numbers are correct.


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

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