
Re: Problem from Willard's _General Topology_
Posted:
Jan 10, 2014 11:35 AM


On 01/05/2014 11:55 AM, Brian M. Scott wrote: > On Sun, 05 Jan 2014 11:09:10 0600, "Michael F. Stemper" > <michael.stemper@gmail.com> wrote in > <news:lac3jp$kfd$1@dontemail.me> in alt.math.undergrad:
Hi, Brian! Thanks for taking the time to respond. I'm sorry about the delay in replying, but I've been chewing over your response, and I see what you've done. However, I'd still appreciate your comments on my solution to Part 1. I've broken it down a little bit more, and switched to your notation.
I'll say "<x_1, ..., x_n>" instead of saying "P_n". In that case, my response to Part 1 is:
<x_1> = { {x_1} }
<x_1, ..., x_(n+1)> = <x_1, ..., x_n> U { (U <x_1, ..., x_n>) U {x_(n+1)} }
Letting n=1 gives <x_1, x_2> = <x_1> U { (U <x_1>) U {x_2} } = { { {x_1} } U { (U { {x_1} }) U {x_2} } = { { {x_1} } U { {x_1} U {x_2} } } = { { {x_1} } U { {x_1, x_2} } } = { {x_1}, {x_1, x_2} }
This is the same definition of <x_1, x_2> that Willard gave in 1C.
Letting n=2 gives <x_1, x_2, x_3> = <x_2> U { (U <x_2>) U {x_3} } = { {x_1}, {x_1, x_2} } U { (U { {x_1}, {x_1, x_2} }) U {x_3} } = { {x_1}, {x_1, x_2} } U { {x_1, x_2} U {x_3} } = { {x_1}, {x_1, x_2} } U { {x_1, x_2, x_3 } } = { {x_1}, {x_1, x_2}, {x_1, x_2, x_3 } }
And, in general, <x_1, ..., x_n> = { {x_1}, ..., {x_1, ..., x_n} }
Is this a valid definition of "ordered ntuple"?
 Michael F. Stemper I feel more like I do now than I did when I came in.

