
Re: Problem from Willard's _General Topology_
Posted:
Jan 5, 2014 12:55 PM


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:
> I've started on _General Topology_ by Stephen Willard, and > am having a little difficulty with Problem 1D, "Cartesian > Products".
> Part 1 of this problem reads: > Provide an inductive definition of "the ordered ntuple (x_1, ..., > x_n) of elements x_1, ..., x_n of a set" so that (x_1, ..., x_n) and > (y_1, ..., y_n) are equal iff their coordinates are equal in order, > i.e., iff x_1=y_1, ..., x_n=y_n.
I prefer to use angle brackets for ordered pairs and ordered ntuples generally and will do so here. You need to have done 1C first, so that you have <x,y> = {{x},{x,y}} as your base case (n=2). Then given <x_1, ..., x_n> and an element x_{n+1}, define
<x_1, ..., x_{n+1}> = <<x_1, ..., x_n>, x_{n+1}>.
If you prefer, you can associate to the right: given <x_1, ..., x_n> and an element x_0, define
<x_0, ..., x_n> =<x_0, <x_1, ..., x_n>>.
The first option means that <x_1, x_2, x_3>, for instance, is actually <<x_1, x_2>, x_3>; the second makes it <x_1, <x_2, x_3>> instead.
> My response to this makes use of the shorthand notation that > P_n = (x_1, ..., x_n). For instance, P_3 = (x_1, x_2, x_3).
Notationally this doesn?t make much sense: you?re trying to write a recursive construction as if it were a proof by induction. You don?t have an object P(n) for each n; you simply have the definition of ?ordered ntuple? for each n.
[...]
> (b) seems simple enough, as well. Define the trivial > Cartesian product of X_1 as { {x}  x in X_1 }. Then, let > the Cartesian product X_1 * ... * X_n * X_(n+1) be > defined as the product (X_1 * ... * X_n) * X_(n+1).
Yes, this will work, though it would be better to write
(X_1 x X_2 x ... x X_n) x X_{n+1}
if you don?t have easy access to
(X_1 × X_2 × ... × X_n) × X_{n+1}. This matches the first of my two approaches to defining ordered ntuples. You could also associate the other way, matching the second.
[...]
Brian

