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

 Messages: [ Previous | Next ]
 quasi Posts: 12,067 Registered: 7/15/05
Re: Stone Cech
Posted: Mar 21, 2013 3:10 PM

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

>> >>>>>
>> >>>>> 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.

quasi

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