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‫ رياضة عام تخلفات من ثالثة (نظام جديد) – كلية‬4 :‫الفرقة‬
‫ الفصل الدراسي الثاني‬- ‫ شعبة الرياضيات‬- )‫التربية(عام‬
‫م‬2010-‫م‬2011
2011 -6 -11 :‫تاريخ االمتحان‬
‫نموذج اجابة ورقة كاملة‬
‫ توبولوجي‬:‫المادة‬
– ‫ عمرو سليمان محمود – جامعة بنها‬/‫ الدكتور‬:‫أسم استاذ المادة‬
‫كلية العلوم – قسم الرياضيات‬
:‫األسئلة‬
Solve the following (three) questions.
First Question:
Decide whether each of the following statements is true or false. Justify your
answer (by providing a proof, a counter example, a correction, an interpretation).
Let ( X , ) be a topological space,   the collection of closed sets and A, B  X :
a) A  B  A  B .
b) A is called open domain (or  o -set ) iff A  Ao .

c) A is called G -set iff A   Gi ; Gi   .
i 1
d) Every closed interval [a, b] is an closed set in the natural topology (i.e.
[a, b]   N ).
e) The constant mapping f : ( X , )  (Y ,  ) : x  f ( x)  p is a continuous map.
Second Question:
1- Let X  1,2,3,  : X ,,{1},{2},{1,2} be a topology on X and
A  {1,3}, B  {1,2} . Find the following:
a)   , Ao , A , B , b( A) and A d
b) a base  for 
c)
d)
e)
f)
the neighborhood systems N x for all x  1,2,3
a local base  (x) for all x  1,2,3
the weight W ( X , ) and the density d ( X , ) .
the relative topology  A on the set A .
2- Third Question:
1- Let X  a, b, c, d ,  : X , ,{a},{a, b},{a, b, c} be a topology on X and
Y  x, y, z, w,  : Y , ,{x},{ y},{x, y},{y, z, w} be a topology on Y . Consider a
map f : X  Y defined by f (a )  y f(b)  f(d)  z, f(c)  w and a map
g : X  Y defined by g(a)  g(b)  x, g(c)  z, g(d)  w . Prove that:
a)
b)
c)
d)
the function
the function
the function
the function
f
f
g
g
is continuous,
is not open,
is not continuous.
is not open.
2- Define a topological space. Prove that if X is an infinite set, then the collection
  : G  X | G is finite   is a topology on the set X .
The answer:
First Question:
a) A  B  A  B (True).:
we have

A  B  A A  B  A


  A  B  A  B - - - - - - - - - - - - - - - - - - - (1)
A  B  B A  B  B


A  A
  A B  A  B  AB  A B
B B
 A B  A  B
(since A  B is closed)
From (1) and (2) the prove is complete
b) A is called open domain (or  o -set ) iff A  Ao (False):
Correction
A is called closed domain (or   -set ) iff A  Ao
- - - - - - - -(2)

c) A is called G -set iff A   Gi ; Gi   (False):
i 1
Correction

A is called G -set iff A   Gi ; Gi  
i 1
d) Every closed interval [a, b] is an closed set in the natural topology (i.e.
[a, b]   N ) (True)
Since [a, b]  (a, b)  N
e) The constant mapping
f : ( X , )  (Y ,  ) : x  f ( x)  p
is a continuous map
(True):
Since
 p  G  f 1 (G)  X 
G    
1
 p  G  f (G)   
Second Question:
Let X  1,2,3,  : X ,,{1},{2},{1,2} be a topology on X and
A  {1,3}, B  {1,2} . We have:
a)

  : X ,,{2,3},{1,3},{3}

Ao  {1}

A  {1,3}  A, B  X  B is dense in X

b( A) : A \ Ao  {1,3} \ {1}  {3}

A d : x  X | x  A \ {x} , so we get:




A \ {2}  {1,3}  A \ {1}  {1,3}  {1,3}  2  A \ {2}  2  A d   A d  {3}

A \ {3}  {1}  A \ {3}  {1}  {1,3}  3  A \ {3}  3  A d

A \ {1}  {3}  A \ {1}  {3}  {3}  1  A \ {1}  1  A d
b) a base  for  is  : X ,{1},{2}
c) the neighborhood systems N x for all x  1,2,3 :
N1 : X ,{1},{1,2}
N 2 : X ,{2},{1,2}
N 3 : X ,
d) a local base  (x) for all x  1,2,3 :
 (1) : {1}
 (2) : {2}
 (3) : X 
e) the weight W ( X , ) :   3 and the density d ( X , ) : B  2
f) the relative topology  A  {A,,{1}}
Third Question:
1) - Let X  a, b, c, d ,  : X , ,{a},{a, b},{a, b, c} be a topology on X and
Y  x, y, z, w,  : Y , ,{x},{ y},{x, y},{y, z, w} be a topology on Y . Consider
a map f : X  Y defined by f (a )  y f(b)  f(d)  z, f(c)  w and a map
g : X  Y defined by g(a)  g(b)  x, g(c)  z, g(d)  w . We get::
a)
the function f is continuous:
f 1{Y }  f 1{{ x, y, z, w}}  X   ,
f 1{}     ,
f 1{x}     ,
f 1{ y}  {a}   ,
f 1{x, y}  {a}   ,
f 1{ y, z , w}  X  
Hence, f is continuous
b) the function f is not open:
f is open iff f (G)   G 
f { X }  f {{a, b, c, d }  Y   ,
f {}     ,
f {a}  { y}   ,
f {a, b}  { y, z}  
c)
the function g is not continuous.
g 1{Y }  g 1{{ x, y, z , w}}  X   ,
g 1{}     ,
g 1{x}  {a, b}   ,
g 1{ y}     ,
g 1{x, y}  {a, b}   ,
g 1{ y, z , w}  {c, d }  
Hence, g is not continuous
d)
the function g is not open:
g{ X }  g{{a, b, c, d }  Y   ,
g{}     ,
g{a}  {x}   ,
g{a, b}  {x}   ,
g{a, b, c}  {x, z}  
2) - A topological space is a pair ( X , ) where
 X is a non-empty set,
   P( X ) such that
1  X , 
2  G1 , G2    G1  G 2  

3  {Gi ; i  1,2,...}     Gi  
i1
To prove that   : G  X | G is finite   is a topology on the set X :
    by definition , X    ( finite)  X  
 We have
G1 , G2     G1, G2 are finite
 (G1  G2 )  G1  G  is finite
 G1  G2   

In fact:
{Gi ; i  1,2,...}     Gi is finite i
 


   Gi    Gi is finite
i 1
 i1 



   Gi    
 i1 
‫انتهت اإلجابة‬
– ‫ عمرو سليمان محمود‬/‫الدكتور‬
‫جامعة بنها – كلية العلوم – قسم‬
‫الرياضيات‬
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