Re: Basic Topology Questions
Zass wrote:
1.1 9 states. Let R be te set of reals. Exactly 3 of the following ten
collection of subsets are topologies. Which 3?
i) R, null, and intervals (a,b) for which a and b are real with a < b.
I say no since the union of 2 nonintersecting intervals is not in the
set.
Correct.
ii) R, null, and intervals (-r, r) for r any positive real.
I say yes .. unions and intersections still lead to intervals (-r, r).
U{(-1+1/n, 1-1/n): n in N, n>0}
iii) R, null, and intervals (-r, r) for r any positive rational.
Yes for the same reason as ii.
Even worse: rationals approach reals.
iv) R, null, and intervals [-r, r] for r any positive rational.
Yes for the same reason as ii.
U{[-1+1/n, 1-1/n]: n in N, n>0}
v) R, null, and intervals (-r, r) for r any positive irrational.
Yes for the same reason as ii.
Aargh: irrationals approach rationals
vi) R, null, and intervals [-r, r] for r any positive irrational.
Yes for the same reason as ii.
No
vii) R, null, and intervals [-r, r) for r any positive real.
Yes for the same reason as ii.
[-1+1/n. 1-1/n)
viii) R, null, and intervals (-r, r] for r any positive real.
Yes for the same reason as ii.
No
ix) R, null, and all intervals [-r, r] and (-r, r) for r any positive
real.
Yes because of iv and ii, and intersections of [-r, r] and (-r, r) is
(-r, r), while unions are [-r, r].
OK
x) R, null, and every interval [-n, n] and (-r, r) for n any positive
integer and r any positive real.
Yes because of an argument similar to ix
OK
--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor in Central New Jersey and New York
.
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