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Compactness in a Metric Space  Date: 10/08/2002 at 20:24:03
From: Pham Subject: Compactness/ closed/open  Please help,
d(p,q)=|p-q|. Let E be the set of all P in Q such that 2<P^2<3.
Show  Is E open in Q?  Thank you very much.  Date: 11/20/2002 at
23:21:58 From: Doctor Nitrogen Subject: Re: Compactness/
closed/open  Hi, Pham:  Try to see if this reasoning helps.
Find convergent sequences of rational numbers     P = 1 + a_n, n
= 1, 2, 3, ........,  with a_n = m/n, m, n integers, and m/n
always an element of Q such      sqrt(2) < 1 + a_n < sqrt(3),
and, for P = (1 + a_n) and P^2 = (1 + a_n)^2 also both
converging:     2 < (1 + a_n)^2 < 3 for each n.   Then the set
E:= {All P in Q | 2 < P^2 < 3}  can be shown to exist.   Then E
can be shown to contain all its limit points, meaning E must  be
closed in Q. You will, of course, have to find suitable
candidates  for a_n and m/n.  But also