Re: Basic Topology Questions
- From: "Justin Young" <x_static66@xxxxxxxxxxx>
- Date: Sun, 12 Jun 2005 01:58:45 GMT
"Zass" <jbeasley@xxxxxxxxx> wrote in message
news:1118538917.522355.147050@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> Hello everyone,
>
> I'm trying to brush up on my topology.. I found an excellent online
> book called Topology Without Tears. It can be found here:
>
> http://uob-community.ballarat.edu.au/~smorris/topbookchaps1-8.pdf
>
> Anyways, I've been reading it and doing exercises, but apparently the
> solutions aren't posted anywhere. This is a little frustrating since I
> know I'm getting some of the answers wrong!
>
> Two of the most basic excercises I know I'm getting wrong. I'd love to
> have someone check my answers, since these test the most basic
> definitions of topology.
>
> The questions I'm having problems with are in exercises 1.1 3 and 1.1
> 9.
>
> 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.
>
> ii) R, null, and intervals (-r, r) for r any positive real.
> I say yes .. unions and intersections still lead to intervals (-r, r).
>
> iii) R, null, and intervals (-r, r) for r any positive rational.
> Yes for the same reason as ii.
show that an "irriational interval" can be obtained using union.
>
> iv) R, null, and intervals [-r, r] for r any positive rational.
> Yes for the same reason as ii.
consider the union of [-r, r] for r in (0, 1) and rational.
>
> v) R, null, and intervals (-r, r) for r any positive irrational.
> Yes for the same reason as ii.
let r_n be positive, irrational and r_n -> 1 from below.
what is the union of (-r_n, r_n)?
>
> vi) R, null, and intervals [-r, r] for r any positive irrational.
> Yes for the same reason as ii.
construct an open interval using union..
>
> vii) R, null, and intervals [-r, r) for r any positive real.
> Yes for the same reason as ii.
construct an open interval using the union
>
> viii) R, null, and intervals (-r, r] for r any positive real.
> Yes for the same reason as ii.
same as above
>
> 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].
i think this is a topology, but you need to show more here.
>
> 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
a little more work here as well.
>
> Now on the other hand, as I think about it more, it seems that ii-x all
> have the property that the infinite intersection of all their subsets
> is just 0 .. but that shouldn't matter since we only consider finite
> intersections.
>
> I must clearly be getting something very wrong since only 3 of them are
> topologies but I find that 9 of them are!
i've got 3.
my advice is to be careful, try out as many different constructions as you
can using
unions and finite intersections.
if you see a restrition like "rational endpoints" see if you can use the
allowed set operations
to get an interval with irrational endpoints.
.
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- Basic Topology Questions
- From: Zass
- Basic Topology Questions
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