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)