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Unformatted text preview: Fall 2003 First Midterm 1)Let ( 29 τ , X be a topological space and A a subset of X. (a) ) ( A Bdd x ∈ if and only if... ) ( A Int x ∈ if and only if... (b) Show that . ) ( ) ( φ = ∩ A Int A Bdd 2) Prove or disprove: (a) { } finite is u or u either u c : ℜ ⊂ = τ is a topolgy on ℜ . (b) { } { }{ }{ } { } 2 , 4 , 2 , 1 , 4 , 2 , , 2 , 1 φ τ = is a topology on { } 4 , 2 , 1 = X . 3) Let ( ] { } 4 2 , ∪ = A be a subset of ℜ . Find Bdd(A), Ext(A) and Int(A) in (a) standard topology on ℜ . (b) leftray topology on ℜ . (c) cocountable topolgy on ℜ . 4) Let ( 29 τ , X be a topological space and A a subset of X. (a) Show that ) ( A cl x ∈ if and only if every open subset of x contains a point of A. (b) Find cl(Q) in the standard topology and in the cocountable topology of ℜ . (c) Is Q closed in the standard topology? Is Q closed in the cocountable topology? Why? Fall 2003 Second Midterm (1) If [ ] [ 6 , 4 ) 3 , × = A is a subset of 2 ℜ . Then find: (a) ) ( A Bd in the standard topology. (b) ) ( A Int in the cofinite topology. (c) ) ( A clus in the leftray topology. (2)(a) Let ) , ( τ X be a topological space. Then a collection β of subsets of X is a basis for τ if... (b) Show that the collection β of all open intervals with rational end points is a base for ) , ( s τ ℜ . (c) Let ) , ( X X τ and ) , ( Y Y τ be topological spaces and { } Y X v u v u τ τ β ∈ ∈ × = , : . Show that β is a basis for some topology on Y X × . What is the name of this topology? (3) Let ) , ( X X τ and ) , ( Y Y τ be topological spaces and Y X f → : a function. Then (a) f is said to be continuous at a point X a ∈ if... (b) Show that f is continuous then X v f τ ∈ ) ( 1 for every Y v τ ∈ . (c) Show that if X w f τ ∈ ) ( 1 for every basic open set Y w ⊂ ,then X v f τ ∈ ) ( 1 for every Y v τ ∈ (4) Let ( ] ℜ ⊂ ∞ = , A . On ℜ define the relation: x x λ if A x ∉ and y x λ ) ( A y ∈ 2200 if A x ∈ . (a) Show that λ is an equivalence relation on ℜ . (b) Describe the quotient space toplogy on λ / ℜ . Fall 2003 Final Exam 1) (a) A topological space ( 29 τ , X is called i) T 1 if .... ii) T if ... iii) regular if... iv) normal if ... (b) Show that ( 29 τ , X is T 1 if and only if each point X x ∈ is a closed subset of X. (c) Give an example of a T 0 space which is not T 1 and an example of a T 1 space. (d) Show that each normal T 1 space is regular. 2) Let ( 29 ( 29 Y X Y X f τ τ , , : → be a bijective and { } X x x f x C ∈ = : )) ( , ( . (a) Show that if C is a closed subset of Y X × , then Y is a T 1 space. (b) Show that if f is open and X is T 1 , then Y is also T 1 . 3) If [ ] [ 6 , 4 ) 3 , × = A is a subset of 2 ℜ . Then find: (a) ) ( A Bd in the standard topology....
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 Spring '11
 fezaaslan
 Logic, Topology, Metric space, Topological space, Hausdorff, ray toplogy

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