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 ,then
iii) for .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
={X, }.
={X, ,{a}}.
={X, ,{b}}.
={X, ,{c}}.
={X, ,{a,b}}.
={X, ,{a,c}}.
={X, ,{b,c}}.
={X, ,{a},{a,b}}.
={X, ,{b},{a,b}}.
={X, ,{c},{a,c}}.
={X, ,{c},{c,b}}.
={X, ,{b},{c,b}}.
={X, ,{a},{a,c}}.
={X, ,{a},{b},{a,b}}.
={X, ,{a},{c},{a,c}}.
={X, ,{c},{b},{c,b}}.
={X, ,{a},{a,c} ,{a,b}}
={X, ,{a},{a,c} ,{a,b}}
={X, ,{c},{a,c} ,{c,b}}
={X, ,{b},{b,c} ,{a,b}}.
={X, ,{a},{a,c} ,{a,b},{b} }
={X, ,{a},{a,c} ,{a,b},{c} }
={X, ,{c},{a,c} ,{c,b},{b} }
={X, ,{c},{a,c} ,{c,b},{a} }
={X, ,{b},{b,c} ,{a,b},{a} }.
={X, ,{b},{b,c} ,{a,b},{c} }.
exercise.
= exercise
={X, ,{b},{a},{c},{a,c},{b,c} ,{a,b}}.
Remark:
={X, }.is called an indiscrete topology.
={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.
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}, ={X, ,{a}} and ={X, ,{b}}are two topologies on X but ={ 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} ).
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)= 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)= 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)= A }= { { },{2} }={2} and
Cl (A)= 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).
closure axioms(بديهيات الانغلاق (
1) X =X and = .
2) E is the smallest closed set containing E or ( E E ).
3) E=E if, and only if,E is closed.
4) E =E 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) X =X
2) E is the largest open set contains in E
3) E E.
4) E =E .
5) (A B) =A B .
6) (A B) A B .
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 ={a}, B ={a,c,d},but(A B) ={a,b,c,d,e} A B .={a,c,d}.