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 <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