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Topic: A toy conjecture which may lead to an other breakthrough conjecture - help request
Replies: 9   Last Post: Sep 8, 2013 4:10 AM

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Victor Porton

Posts: 621
Registered: 8/1/05
Re: A toy conjecture which may lead to an other breakthrough conjecture - help request
Posted: Sep 7, 2013 8:48 AM
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William Elliot wrote:

> On Fri, 6 Sep 2013, Victor Porton wrote:
>

>> Please help with a solution. This is VERY important for development of
>> general topology.

>
> Why is it important for topology?


I was expiring to prove this conjecture:

http://portonmath.wordpress.com/2013/09/01/funcoids-open-mappings/

I proved this important conjecture yesterday.

This conjecture is an analog for funcoids of the statement that in Top there
are direct products. It is a very important statement.

>> Let A be a set.
>> Let pi_0, pi_1 be projections from AxA.
>> Let F_0, F_1, G_0, G_1 be binary relations on A.

>
>> Let phi_A be the maximal binary relation in (AxA) x (AxA) such that
>> pi_0 o phi_A subset F_0 o pi_0 and pi_1 o phi_A subset F_1 o pi_1.

>
> Consistent with F_0 being a binary relation on AxA don't you mean
> "on"? Indeed, a binary relation on A is not "in" AxA but a subset
> of AxA.


It seems that your grammar notes are true. It is not important anymore
anyway after I've proved a more general conjecture.

> Is phi_A the unique maximal relation as implied by the use of
> "the" (making phi_A a maximum) or to you mean "_a_ maximal".


In the lattice of binary relations it cannot be more than one maximum.

> In what font size is your pdf book? Can you change it?
> What fort sizes can you use? Can Acrobat reader print
> a range pages without printing the whole file?


10pt.

It is not very hard for me to create a version for you with a different font
size. (Which size do you want?) But some formulas may probably overflow
outside of the paper.

Acrobat Reader can print a range of pages, but only in PostScript (not PDF)
format. My Acrobat Reader 9.5.5 for Linux has create an erroneous PostScript
file which cannot be read.

To extract a range of pages from a PDF file to an other PDF file, I use
"Print" menu item in Okular (probably not available on Windows). You may
also try Evince (maybe available for Windows).

>> Prove (or disprove) that Sigma = phi_B o phi_A is the maximal
>> binary relation on A such that pi_0 o Sigma subset G_0 o
>> F_0 o pi_0 and pi_1 o Sigma subset G_1 o F_1 o pi_1.





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