Download Topological spaces and basic definitions Introduction: In this lecture

Lecture 1 : Topology
Lecture : K . Al-Ghurabi
Topological spaces and basic definitions
Introduction:
In this lecture we shall try to investigate some basic definitions in a
topological spaces, so what is the topology and what is the types of
topologies and what is the main concepts in topology ,this lecture will
answer all these questions
Definition:
Let X be a non-empty set and let  be a collection of a subset of X
satisfying the following conditions:
i) X,   
ii) if G1 ,G2  ,then G1  G2 
iii)  Gi   for    .i  I

then  is called a topology on X and (X,  ) is a topological space and for
easy written by X.
Remark: the elements of is called open sets .
Example:
Let X={a,b,c},  ={X,  ,{a} }. Is topology on X, since  satisfying the
conditions (i),(ii) and (iii).
Examples:
Let X={a,b,c} ,then
 1 ={X,  }.
 2 ={X,  ,{a}}.
 3 ={X,  ,{b}}.
 4 ={X,  ,{c}}.
 5 ={X,  ,{a,b}}.
 6 ={X,  ,{a,c}}.
 7 ={X,  ,{b,c}}.
 8 ={X,  ,{a},{a,b}}.
 9 ={X,  ,{b},{a,b}}.
 10 ={X,  ,{c},{a,c}}.
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Lecture 1 : Topology
Lecture : K . Al-Ghurabi
11={X,  ,{c},{c,b}}.
 12 ={X,  ,{b},{c,b}}.
 13 ={X,  ,{a},{a,c}}.
 14 ={X,  ,{a},{b},{a,b}}.
 15 ={X,  ,{a},{c},{a,c}}.
 16 ={X,  ,{c},{b},{c,b}}.
 17 ={X,  ,{a},{a,c} ,{a,b}}
 18 ={X,  ,{a},{a,c} ,{a,b}}
 19 ={X,  ,{c},{a,c} ,{c,b}}
 20 ={X,  ,{b},{b,c} ,{a,b}}.
 21 ={X,  ,{a},{a,c} ,{a,b},{b} }
 22 ={X,  ,{a},{a,c} ,{a,b},{c} }
 23 ={X,  ,{c},{a,c} ,{c,b},{b} }
 24 ={X,  ,{c},{a,c} ,{c,b},{a} }
 25 ={X,  ,{b},{b,c} ,{a,b},{a} }.
 26 ={X,  ,{b},{b,c} ,{a,b},{c} }.
 27  exercise.
 28 = exercise
 29 ={X,  ,{b},{a},{c},{a,c},{b,c} ,{a,b}}.
Remark:
 1 ={X,  }.is called an indiscrete topology.
 29 ={X,  ,{b},{a},{c},{a,c},{b,c} ,{a,b}}. Is called a discrete topology.
Example: letX={a,b,c},  ={X,  ,{a,c} ,{c,b},{a} },then  is not topology
on X, since {a,c}  {c,b}={c} 
Example: letX={a,b,c},  ={X,  ,{a,c} ,{c,b},{b} },then  is not
topology on X, since {a,c}  {c,b}={c} 
Example: letX={a,b,c},  ={X,  ,{a,c} ,{a,b},{c} },then  is not topology
on X, since {a,c}  {a,b}={a} 
Example: letX={a,b,c},  ={X,  ,{a,c} ,{c,b},{a},{b} },then  is not
topology on X, since {a,c}  {c,b}={c}  and {a}  {b}={a,b}
Example: letX={a,b,c},  ={X, ,{a,c} ,{c,b},{a},{b} },then  is not
topology on X, since    .
Exercise: let X={1,2,3},then find a topologies on X.
II
Lecture 1 : Topology
Lecture : K . Al-Ghurabi
Example: let X=N, where N is the set of natural numbers then
,C={G  N:X/G is finite} is a topology on X ,and it is called co finite
topology
And prove that C is a topology on X=N.
Also if V={G  N:X/G is countable} is a topology on X and it is called
co countable topology on X ( prove that ).
Exercise: let X={a,b,c,d},then find a topologies on X.
Theorem: the intersection of a family of topologies defined on the same
set is a topology on this set.
remark: the union of a family of topologies defined on the same set need
not be a topology on this set
Example: let X={a,b,c},  1 ={X,  ,{a}} and  2 ={X,  ,{b}}are two
topologies on X but  1   2 ={ X,  ,{a},{b}} is not topology on X .
limit points (‫) نقاط الغايه‬
Definition: let A be a sub set of a topological space (X,  ) and x  X , we
say that x is a limit point of A if ,and only if, for any open set G such that
x  G, A  G-{x}   , and the set of all limit points of A is called (derived
set) and it is denoted by d(A).
Example: let X={a,b,c,d,e}, ={ X,  ,{a},{b,d},{a,b,d},{b,c,d,e} and
A={b,c,d},B={a,b,c},then find d(A),d(B).
Solution:
a  {a},a  {a,b,d},a  X, now let G={a},then A  {a}-{a}={a,b,c}  {a}{a}=  ,so x  d(A)
b  {b,d},b  {a,b,d},b  {b,c,d,e},b  X, now let G={b,d},then A  G{b}={a,b,c}  {b,d}-{b}=  ,so b  d(A)……………etc.
and if continue we shall find that d(A)={c,d,e} and that d(B)={b,c,d,e}.
Proposition: let A,B,E are sub sets of a topological space(X,  ) ,then
1) d(  )=  .
2) If A  B then d( A)  d(B).
3) d(A  B)=d(A)  d(B).
4) if x  d(A) then x  d(A-{x} ).
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Lecture 1 : Topology
Lecture : K . Al-Ghurabi
Open and closed sets (‫) المجموعات المغلقة والمجموعات المفتوحة‬
Definition: let A be a sub set of a topological space (X,  ),the interior of
A is defined by:
Int (A)=  {G   : G  A } or it is the largest open set contains in A.
Definition: let A be a sub set of a topological space (X,  ),the closure of
A is defined by:
Cl (A)=  { X / G   : G  A } or it is the smallest closed set containing A.
Example: let X={1,2,3},  ={ X,  ,{1},{1,3} ,{1,2},{2} },and let
A={2,3}
Then find int(A) and cl(A).
Solution:
Int (A)=  {G   : G  A }=  { {  },{2} }={2} and
Cl (A)=  { X / G   : G  A }=  { {2,3} ,X }={2,3}.
Definition: a sub set A of a topological space (X,  ) is said to be open if
X/A is closed.
Definition: a sub set A of a topological space (X,  ) is said to be closed if
X/A is open.
Theorem: a sub set A of a topological space (X,  ) is open if, and only if
A=int(A).
Theorem: a sub set A of a topological space (X,  ) is closed if, and onlyif
A=cl(A).
Example: le tX={a,b,c},  ={X,  ,{a,c} ,{c,b},{a} },then the open sets are
X,  ,{a,c} ,{c,b},{a} so by theorem above we have
int(X)=X,int{a,c}={a,c}, int{c,b}={c,b},int{a}={a}
The closed sets are  , X,{b} ,{a},{b,c},then by theorem above we have
cl(X)=X,cl{b}={b},cl{a}={a},cl{b,c}={b,c}.
Proposition: let E be a sub set of a topological space(X, ) , then
cl(E)=E  d(E).
IV
Lecture 1 : Topology
Lecture : K . Al-Ghurabi
closure axioms(‫( بديهيات االنغالق‬
__
__
=X and  =  .
1) X
__
__
2) E is the smallest closed set containing E or ( E  E ).
__
3) E=E
__
4) E =E
if, and only if,E is closed.
__ __
or cl(E)=cl(cl(E)).
5) cl(A  B)= cl(A)  cl(B).
6) cl(A  B)  cl(A)  cl(B).
Example on (6)
let X={a,b,c,d,e},  ={X,  ,{a},{b,c},{a,b,c},{a,b,c,d}
},A={a,b,d},B={c,d},A  B={d},cl(A)=X,cl(B)={b,c,d,e} then
cl(A  B)={d,e}, cl(A)  cl(B)={b,c,d,e},hence
cl(A  B) 
cl(A)  cl(B).
Interior axioms )‫) البديهيات الداخلية‬
1)
2)
3)
4)
5)
6)
O
X =X
E O is the largest open set contains in E
E O  E.
E O O =E O .
(A  B) O =A O  B O .
(A  B) O  A O  B O .
Example on (6).
LetX={a,b,c,d,e}  ={X,  ,{a},{c,d},{a,c,d},{b,c,d,e} } and let
A={a,b,e},B={a,c,d},A  B={a,b,c,d,e}
Now
A O ={a}, B O ={a,c,d},but(A  B) O ={a,b,c,d,e}  A O  B O .={a,c,d}.
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