From: hrubin@pop.stat.purdue.edu (Herman Rubin)
Newsgroups: sci.math
Subject: Re: The compact-open topology
Date: 9 Jul 92 15:00:27 GMT

In article <1992Jul8.142804.17523@galois.mit.edu> jbaez@cayley.mit.edu (John C. Baez) writes:
>In article <1992Jul8.005210.2384@athena.cas.vanderbilt.edu> rickertj@athena.cas.vanderbilt.edu.UUCP (John Rickert) writes:

>>   Hello.  I'm looking for some help concerning the compact-open
>>topology.  I've stared at the definition a few times, but it still
>>doesn't make much sense.  Is there a way to "visualize" the topology?
>>Is there some good motivation for the definition?  How important is
>>it to know?  I'm fishing for an answer like, "You need to know it,"
>>or, "It's a good thing to know if..." or, "Have a nodding acquaintance
>>with it."

>I'm probably not the one to answer this, since of all the topologies on
>functions I've used (uniform, pointwise, pointwise a.e., L^p, C^k,
>C^infty, Schwartz, W^{p,q}, etc.) this is one I've never used.  You may
>conclude that it's a useless topology, but I think if I were a
>topologist rather than an analyst I would have used this one more.  It
>seems like a fine upstanding sort of topology to me, in any event.
>Think of it as a slight sharpening of pointwise convergence, if you like.

It is quite useful in analysis as well as topology.  As far as visualizing
it, the definition is not that unclear.  Consider a compact set in the
topology of the argument, and any neighborhood of the range of that compact
set.  Then those functions which map the compact set into that neighborhood
form a neighborhood in the c-o topology.

The importance is that it is what is called an admissible topology for
continuous functions, and for locally compact spaces, the weakest one.
That is, if f_a converges to g, and x_a converges to y, then 
f_a(x_a) converges to g(y).  It is the appropriate generalization
of uniform convergence to non-compact locally compact spaces.
-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@pop.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!pop.stat!hrubin(UUCP)