M.Sc .Taha Guma el turki  , Prof. Al mabrouk Ali sola 


There was a beautiful mathematical work done by Kherie Mohamed mera & Prof.Al mabrouk Ali sola .Related to extremal topologies and how to extract the extremal topologies and their  numbers by a formula . 

Definition :-

Let X be a set and ,T is not a discreet topology on X then T is said to be an extremal topology if every topology strictly finer than T is discreet.

Theorem 1-2 of [1] :- 

 If  X is any set with more than one element , x , y ∈ X , x ≠y , and
T_{x,y}= P(X\{x}) U {{x} U A , A ∈P(X\{x}),y ∈ A} ,then T_{x,y} is
an extremal topology on X [1] .

Remark

i-Notice that if X is a set  x,y ∈ X , x≠ y , then T_{x,y} ≠T_{y,x} [1].

Theorem  2-1 of [1]  :-

    Any extremal topology on a finite set with more than one element is in the form T_{x,y} for some x,y ∈ X , x ≠y [1] .

Theorem 2-2 of [1] :-

If X is a set has n elements then the number of extremal topologies defined on X is n(n-1) [1].

Theorem 2-3 of [1]  :-

If  X is a set with n elements then any extremal topology has 3(2^n-2) elements [1] .

Also we compute in this application the Non-trivial minimal topologies and there number  which is equal to 2ⁿ-2 ;

Notes :-

1- The  Author of the procedures: Taha Guma el turki uses low speed computer  with 1.7 GH processor.

2- If you use such or lower portable then replace ; by : at the end of procedure calling To compute issues for  n>10.

3- The users can easily remove #Example(2) and #Example(3) and use the application for arbitrary n depending  on their computer options .

 References

[1] A lmabrouk Ali Sola , Extremal Topologies ,Damascus University Journal of BASIC SCIENCES,2005,Vol.21,No 1,19-25 .