Re: Open Sets & Subspace Topologies

In article <16436630.1128018318895.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Narcoleptic Insomniac <i_have_narcoleptic_insomnia@xxxxxxxxx> wrote:
>**This is a homework exercise so only hints please**
>
>If anyone could take a look at my reasoning and let me
>know if this seems correct I'd appreciate it.
>
>Consider the subspace Y = [-1, 1] of R where R has the
>usual topology T. Then the subspace topology T_y is
>defined as the collection {[-1, 1] /\ K : K is in T}.
>It's clear to me that it is possible for a set to be
>open in Y but not be open in R.
>
>Okay, now let C be the set C = {x : 1/2 <= |x| < 1} which
>can be written as C = (-1, -1/2] \/ [1/2, 1). I know
>that C is not open in R since C is not an element of T
>(that is to say, C cannot be expressed as a union nor as
>an intersection of open sets in R). Moreover, C is not
>open in Y either, since it is not in the collection T_y.

Well, surely you would have to justify that last part. How do you know
there is no weird open set in R which when you intersect it with Y
just happens to yields C?

>Now consider the set B = {x : 1/2 < |x| <= 1} which can
>expressed as B = [-1, -1/2) \/ (1/2, 1]. This set is not
>open in T for the same reasons that C was not open in T.
>However, B is open in T_y since the set [-1, -1/2) can
>be seen as the intersection [-1, 1] /\ (-n, -1/2) where
>n > 1. The argument is similar for (1/2, 1] and thus the
>union (which is B) is open in T_y.
>
>These aren't formal proofs,

I'm unclear what statement you are trying to establish in any
case. Are you trying to show that it is possible for a subset of Y to
be open in Y, but not open when you consider it a subspace of R? Or
somethign else? Are you trying to prove that B is open in Y but not in
R, and that C is not open in either R or Y?

> but I just wanted to see if
>the reasoning and logic I'm using seems correct.

If you are trying to show that B is open in Y but not in R, then I
would suggest using a specific value of n rather than your unspecified
n above; just exhibit the open sets you want, explicitly and with
nothing left in the air. If you are also trying to show that C is not
open in Y, then I would say you have a bit more work to do. You need
to explain why there is no open set U in R such that U /\ Y = C. You
should be able to do this easily, but noting what the fact that 1/2
and -1/2 have to be in U implies for the intersection.


--
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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
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Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.