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

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Victor Porton

Posts: 621
Registered: 8/1/05
Re: About generalizations
Posted: Aug 15, 2013 4:40 PM
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FredJeffries wrote:

> On Thursday, August 15, 2013 6:17:49 AM UTC-7, 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 endo-reloids. So by abuse of notation
we will consider mu and nu as endo-reloids.

Let F is the reloid induced by the function f.

The uniform continuity in the exercise's conditions is expressed by the

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:

Victor Porton -

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