‫بسم هللا الرحمن الرحيم‬
Kingdom of Saudi Arabia
‫الـمـمـلكـة الـعـربـيـة الـسـعـوديـة‬
‫وزارة الـتـعـلـيـم الـعـالـي‬
‫جـامـعـة الـمـجـمـعـة‬
‫كلية العلوم بالزلفي‬
Ministry of Higher Education
Majmaah University
College Of Sciences in Alzulfi
non -:‫تاريخ االمتحان‬
‫ السادس‬-:‫المستوى‬
MAT 373-Z -:‫رقم ورمز المادة‬
‫ مقدمة في التو بولوجي‬-:‫اسم المادة‬
Answer the Following Questions:
Q1) a) Define a compact space and prove if ( X , ) is T2  space and F is compact set
In X then F is closed
b) Prove: A topological space ( X , ) is T1  space iff x  X , and then x is a closed set
c) Define T0 , hereditary property and prove T0 space is a hereditary property
Q2) a) Define T2  space and prove that every discrete space is not compact and indiscrete space is
compact
b) Consider the topology 𝜏 = {𝑋, ∅, {1}, {1,4}, {1,3,4}} on 𝑋 = {1,2,3,4} and let 𝐴 = {2,3,4}.
Find 𝐴° , 𝐴 , 𝑏(𝐴), 𝐴́
c) Define homeomorphism, topological property and prove that the density is a topological property
Q3) a) correct the following statements (if it’s incorrect)
(any statement mark)
1)
Let 𝑋 = {𝑎, 𝑏, 𝑐}, 𝜏 = {𝑋, ∅, {𝑎}, {𝑏}, {𝑎, 𝑐}} . Then 𝜏 is a topology on 𝑋
2)
Let 𝑋 = {1,2,3}, 𝜏 = {𝑋, ∅, {1}, {1,2}} is a topology on 𝑋, if 𝐴 = {2,3}. Then 𝐴° = {2,3}
3)
A topological space (𝑋, 𝜏) is 𝑇1 -space if ∀𝑥, 𝑦 ∈ 𝑋, ∃𝑢, 𝑣 ∈ 𝜏 𝑠. 𝑡. , 𝑥 ∈ 𝑢, 𝑦 ∈ 𝑣
4)
A topological space (𝑋, 𝜏) is compact space if every open cover has a finite sub cover
5)
Let 𝑋 = {𝛼, 𝛽, 𝛾, 𝜀, 𝜖, 𝜃}, 𝜏 = {𝑋, ∅, {𝛽}, {𝛽, 𝜀}} is a topology on 𝑋. Then (𝑋, 𝜏) is connected space
6)
If ( X , ) is a topological space, A  X then A'  A is open set
7)
If ( X , ) is open sets a topological space, A  X then A open sets, A  closed sets,
A'  A,
b( A)  A  A
8)
In usual topology ( R, U ) , (0,1) '  [0,1)
9)
If ( X , ) is a topological space, A  X then  A  { :   } is topology on A
1
‫بسم هللا الرحمن الرحيم‬
Kingdom of Saudi Arabia
‫الـمـمـلكـة الـعـربـيـة الـسـعـوديـة‬
‫وزارة الـتـعـلـيـم الـعـالـي‬
‫جـامـعـة الـمـجـمـعـة‬
‫كلية العلوم بالزلفي‬
Ministry of Higher Education
Majmaah University
College Of Sciences in Alzulfi
non -:‫تاريخ االمتحان‬
‫ السادس‬-:‫المستوى‬
MAT 373-Z -:‫رقم ورمز المادة‬
‫ مقدمة في التو بولوجي‬-:‫اسم المادة‬
10) If ( X , ) is a topological space, P  X then  is a neighborhood of a point P if
 G 
s.t. P  G  X
11) If  is a base of a topological space ( X , ) , then    Bi ,
i
Bi  
12) If ( X , I ) is indiscrete topological space, ( X , D ) is a discrete topological space, then
f : ( X , I )  ( X , D) is continuous function
𝑏) Define a complete lattice, linear order and prove A subset A of a topological space X is closed iff A
contains each of its accumulation point
c) Define continuity and Prove: A function f : X  Y is continuous if and only if the inverse image of every
closed subsets of Y is closed subsets of X
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