Date: Thu, 05 Jun 1997 23:51:24 +1000
From: [Permisson pending]
To: rusin@washington.math.niu.edu
Subject: in need of topological help
[deletia -- djr]
I have a curly question that I am stuck on -
could you please help me?
Let X nonempty, and (Y,s) be a topological space.
Let {fi}, i in I be a family of functions with fi : X -> Y.
Let t be the smallest topology on X : all functions fi are
continuous (I can explain why this exists).
If {x(a)}, a in A is a net in X, I can show that
x(a) -> x in the t topology implies fi(x(a)) -> fi(x) for all i in I
(easy)
I have been trying to prove the converse, ie that
fi(x(a)) -> fi(x) for all i in I implies x(a) -> x in the t topology
without success. Can you help me?
I start by assuming fi(x(a)) -> fi(x) for all i in I, of course.
Then we go via contradiction ...
Suppose that x(a) does not -> x in the t topology,
ie suppose that
there exists a U in t with x in U for all numbers N such that
there is an n > N with x(n) not in U.
{obviously I want to continue until I reach a contradiction of the
continuity of the fi's.}
By assumption [fi(x(a)) -> fi(x) for all i in I],
for all V in s such that fi(x) is in V,
there exists an M > 0 :
m > M implies fi(x(m)) is in V.
Then to gain the contradiction I think that I should show that
f(inverse)i(V) is not in t.
Here I am stuck.
If you can help me, I wait with many thanks.
If my notation is too awkward or the question unclear, could you
please write back?
If you are too busy, I understand. Please reply with a blank email.
Thankyou so much for all of your time
[Permission pending]
==============================================================================
Date: Thu, 5 Jun 1997 10:51:59 -0500 (CDT)
From: Dave Rusin
To: mufasa@ozemail.com.au
Subject: Re: in need of topological help
>Let X nonempty, and (Y,s) be a topological space.
>Let {fi}, i in I be a family of functions with fi : X -> Y.
>Let t be the smallest topology on X : all functions fi are
>continuous (I can explain why this exists).
>
>If {x(a)}, a in A is a net in X, I can show that
>
>x(a) -> x in the t topology implies fi(x(a)) -> fi(x) for all i in I
>(easy)
>
>I have been trying to prove the converse, ie that
>
>fi(x(a)) -> fi(x) for all i in I implies x(a) -> x in the t topology
>
>without success. Can you help me?
Perhaps we have different terminology here but this seems straightforward.
Your topology on X will consist of unions of finite intersections of
sets of the form V_i = f_i^(-1)(U_i), U_i open in Y. In particular,
any open set in X around x contains a finite intersection of such
sets V_i where the corresponding U_i contain f_i(x).
For each such i, if f_i(x(a)) -> f_i(x), then for all but finitely
many a we must have f_i(x(a)) in U_i, so that x(a) lies in V_i for
all but finitely many a. Thus x(a) lies in the intersection of all
these V_i except for a finite union of finite sets of a's.
dave