Lecture 2: Topology
Bases and sub bases (‫) اﻟﻘﻮاﻋﺪ و اﻟﻘﻮاﻋﺪ اﻟﺠﺰءﯾﮫ‬
Introduction:
In the last lecture we studied and defined a topological spaces and them
types, also we studied and investigate many facts about these spaces, also
we referred to the derived sets and we knew that the family of open sets is
enough to build the topology .
In this lecture we shall know by studding a family smallest than the
family of the open sets which it is enough to define a topology on the set,
for example τ={Φ,X ,{a},{b},{a,b} } on X={a,b,c}is determined by
know the elements which is contained in it Φ,X ,{a},{b},but this
topology can be determined without {a,b},since a topology will contained
it ,and by the same method the discrete topology can be determined, all
this can be lead us to try know the enough sub set to determined a
topology on set and this is the goal of this lecture.
The bases
Definition:
Let (X, τ) be a topological space a nonempty collection B(x) of T-nbd of
x is called a base for the Tnbd system of x iff for every T-nbd N of x
there is BXB(x) such that BTN.
Remark:
A definition above is equivalent that    is abase for τ if and only if for
all p  G such that G  τ, there exist B   such that p  B  G .
Example:
Let X={a,b,c,d} and  ={ {a,b},{b,c } } and let  is abase for a topology τ
on X then must be that the elements {a,b},{b,c} in τ so we must have
{a,b}  {b,c}={b} τ but {b} can not be expressed as a union of elments
of  ,this help us to know that the choice of  must have this condition:
If A,B   implies that A  B=  {B:B   }.
Let  be a family of a sub set of X ,then  is a base for a topology on X
if and only if the following conditions are hold:
i)
X=  {B:B   },
I
ii)
For any B1 , B2   , we can expressed a bout B1  B2 as a union
of members of  .
Definition:
Let (X, τ) be a topological space and S  τ is said to be sub base for τ
If and only if the finite intersection of members of S is a base for τ.
Example:
Let X={a,b,c,d,e} and S={X,{a,b,c},{c,d},{d,e} }then the family of
all
the
intersections
of
members
of
S
is
 ={
X,{a,b,c},{c,d},{d,e},{c},{d}, Φ }, and the family of union of
members of  gives the topology τ = {X,{a,b,c},{c,d},{d,e},{c},{d},
Φ,{c,d,e},{a,b,c,d} }.
Example:
Let X={a,b,c} and S={X,{a},{b} },then find a topology on X.
Solution:
 ={ X,{a},{b}, Φ } and τ={ X,{a},{b}, Φ,{a,b} }.
Example:
Let X={a,b,c} and S={X,{a}} ,then  = τ={ X,{a}, Φ }.
Example:
Let X={1,2,3} and S={X, {1},{2,3},{2} }, then  ={ X, {1},{2,3},{2}
Φ}, τ={ X, {1},{2,3},{2}, Φ,{1,2} }.
exercises:
Let X={a,b,c} and if
i)
S={X,{b} } then find a topology on X.
ii)
S={ X,{a},{c} } then find a topology on X.
iii)
S={X, {a},{b,c},{b} } then find a topology on X.
Theorem:
Let (X,T) be a topological space and let B(x) be a local base at any point
x of X then B(x) has the following properties:
1- B(x)≠Ø for every xXX
2- If BXB(x) then xXB
II
3- If AXB(x) and BXB(x) then there exist a CXB(x) such that
CTAWB
Proof: discussed in the lecture
limit points (‫) ﻧﻘﺎط اﻟﻐﺎﯾﮫ‬
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) D(AWB)=D(A)WD(B)
Proof: discussed in the lecture
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}.
Theorem: a sub set A of a topological space (X, ) is open if, and only if
III
A=int(A).
Proof: discussed in the lecture
Theorem: a sub set A of a topological space (X, ) is closed if, and onlyif
A=cl(A).
Proof: discussed in the lecture
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).
Proof: discussed in the lecture
closure axioms(‫) ﺑﺪﯾﮭﯿﺎت اﻻﻧﻐﻼق‬
1) Cl(X) =X and cl(Ø)=Ø.
2) Cl(E) is the smallest closed set containing E or ( E  cl(E)).
3) E=cl(E) if, and only if,E is closed.
4) cl(E)=cl(cl(E)).
5) cl(A  B)= cl(A)  cl(B).
6) cl(A  B)  cl(A)  cl(B).
proof: discussed in the lecture
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 (‫) اﻟﺒﺪﯾﮭﯿﺎت اﻟﺪاﺧﻠﯿﺔ‬
O
1) X =X
O
2) E is the largest open set contains in E
O
3) E  E.
O O
O
4) E
=E .
O
O
O
5) (A  B) =A  B .
6) (A  B)
O
 AO  BO .
Proof: discussed in the lecture
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}
IV
Now
O
O
O
O
O
A ={a}, B ={a,c,d},but(A  B) ={a,b,c,d,e}  A  B .={a,c,d}.
V