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Topic:
About generalizations
Replies:
21
Last Post:
Aug 20, 2013 11:45 PM




Re: About generalizations
Posted:
Aug 15, 2013 4:40 PM


FredJeffries wrote:
> On Thursday, August 15, 2013 6:17:49 AM UTC7, Victor Porton wrote: >> >> But it does not make sense to speak in short simple sentences about my >> abstract research. > > What problems can you solve using your methods/techniques? > > I'm not talking about unsolved problems. Just take an exercise from > a topology textbook and show us how to solve it.
Exercise. Prove that uniformly continuous function is proximally continuous regarding the proximity induced by a given uniformity.
Proof. Let f is a uniformly continuous function from a uniform space mu to a uniform space nu.
Uniform spaces are essentially certain endoreloids. So by abuse of notation we will consider mu and nu as endoreloids.
Let F is the reloid induced by the function f.
The uniform continuity in the exercise's conditions is expressed by the formula
F o mu <= nu o F.
Apply the map (FCD) (the funcoid corresponding to a reloid) to this formula, taking into account that (FCD) is distributive over composition of reloids:
(FCD)F o (FCD)mu <= (FCD)nu o (FCD)F.
Taking into account that (FCD)F is equal to the principal funcoid induced by f, we conclude that f is a proximally continuous function from (FCD)mu to the (FCD)nu.
The terminology above is borrowed from my book:
http://www.mathematics21.org/algebraicgeneraltopology.html
 Victor Porton  http://portonvictor.org



