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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) Gi 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}}. I 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} ). III 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}. V