課程名稱︰分析一 課程性質︰數學系選修 可抵數學系必修分析導論一 課程教師︰齊震宇教授 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰第一部分: 105.12.13, 第二部分: 105.12.13 ~ 12.15 考試時限（分鐘）：第一部分: 3 hours, 第二部分: 2 days 試題 : 第一部分 1. STATEMENT OF DEFINITIONS AND THEOREMS (1 POINT EACH) 1. Let F be a family of maps from a metric space (X, d ) to another (Y, d ). X Y For a point x in X, we say that F is equicontinuous at x if 2. A poset (X, ≧) is well-ordered if 3. Let X be a topological space and A is a subset of X. A point x in X is a boundary point of A if 4. Let X be a topological space and A is a subset of X. A point x in X is a limit point of A if 5. Let X be a topological space. B, a subset of P(X), is a local basis of X at a point x in X if 6. A topological space is Lindelöf if 7. A topologocal space is regular(T_3) if x. 8. A net (D, ≧) → X in a topological space X converges to a point x in X if h 9. Let (D, ≧) and (D', ≧') be two directed sets. A map D' → D is final with respect to ≧ and ≧' if x. 10. A net (D, ≧) → X lies in a subset A of X frequently if x. 11. A net (D, ≧) → X lies in a subset A of X eventually if 12. A family F of subsets of a set X has the finite intersection property if f 13. A map X → Y between topological spaces is a quotient map if 14. State Zorn's lemma. 15. State the Hahn-Banach theorem. n In the following, unless the contrary is stated, we always equip R with the n topology induced by the standard Euclidean metric and equip every subset of R with the subspace topology. 2. TRUE AND FALSE (2 POINTS EACH) 1. There is a set which has exactly one topology. 2. There is a set which has exactly two different topologies. 3. There is a set which has exactly three different topologies. 4. There is a set which has exactly four different topologies. 5. There is a set which has exactly five different topologies. 6. There is a set which has infinitely many different topologies. 7. Let X be a locally compact Hausdorff space, C a closed subset and K a compact set of X with C∩K empty. There exists open neighborhoods U and V of C and K respectively with U∩V empty. 8. Every metric space is normal. f 2 9. If (0, 1) → R is a continuous injection, then its image is not compact. 10. A compact subset K in a topological space X is also a closed subset of X. 11. If F is a sequence of convex functions on [0,1] which are uniformly n bounded, then F admits a uniformly convergent subsequence. n 3. PROOF 1. (10 points) Let X be a separable metric space and F a family of R-valued functions on X such that (1) F is equicontinuous at every x in X and (2) for every x in X the set {f(x)|f in F} is bounded. Then every sequence f in F n admits a subsequence which converges pointwise. 2. (10 points) Given two sets A and B, there exists an injection from A to B or from B to A. 3. (10 points) Does there exist a uncountable set X which has an R-valued non-constant continuous function on X when equipped with the cocountable topology? If your answer is yes, provide an example which should consist of an X and a non-constant continuous function, and justify it; if no, prove that it is never possible. 4. (10 points) If X is a totally bounded and complete metric space then X is compact. 5. (10 points) If x (α in D) is a universal net in a compact space X then α it converges. fn 6. (10 points) Let X → Y be a sequence of maps from a topologocal space X to a metric space Y which is continuous at a specific point x_0 in X. If fn converges uniformly to a map f as n tends to infinity, then f is also continuous at x_0. 7. (10 points) Let X be a topological space and A a subset of X. _ (1) (5 points) A = {x in X|there exists a net in A converging to x} x. (2) (10 points) Let (D, ≧) → X be a net. Then the net lies in A frequently if and only if it has a subset lying in A eventually. 第二部分 1. (15 points) Let X be a separable metric space and F a family of R-valued functions on X such that (1) F is equicontinuous at every x in X and (2) for every x in X the set {f(x)|f in F} is bounded. Is it true that every sequence fn in F admits a subsequence which converges compactly? If your answer is yes, prove it; if no, find a counterexample. (Note that this statement is stronger than the generalization of Ascoli's theorem in Exercise 1.3 of the notes I gave you since we remove the condition (0) there.) 2. (10 points) Let X be a metric space and Z be a subset of X. Consider the following two statements: (1) For any given ε＞ 0, there exists a finite subset S of X such that Z is covered by ∪ B (s). s in S ε (2) For any given ε＞ 0, there exists a finite subset S of Z such that Z is covered by ∪ B (s). s in S ε Show that (1) implies (2). ((2) implies (1) obviously.) f 3. Let X → Y be a map between topological spaces. Suppose that X (j in J) is j a family of subspaces of X such that X = ∪ X and f∣ is continuous on j in J j ∣X j X for every j in J. Show that f is continuous if either j (1) (3 points) X is open in X for every j in J. j (2) (7 points) X (j in J) is a locally finite family of closed sibsets in X. j 2 4. (15 points) Let P = {(x,y) in R | |x| ≦ 2, |x + y| ≦ 3, |x - y| ≦ 3} and 3 let Y be a surface of 2 "handles" in R as shown by the following picture. 2 3 We equip P and Y with the subspace topologies inherited from R and R f respectively. Suppose that P → Y is a continuous surjection and let R be the equivalence relation on P such that for p, p' in P, pRp' if and only if f(p) = f(p'). Show that there is a homeomorphism from P/R (equipped with the quotient topology) onto Y. p 5. (15 points) Let X → Y be a quotient map between topological spaces. Equip X ×R and Y ×R with the product topologies. Show that the map q X ×R → Y ×R is also a quotient map. (x, t)→ (p(x), t) 7. Let C[z , z ] be the set of all polynomials with complex coefficients in 1 2 two variables z and z . For any S, subset of C[z , z ] we define 1 2 1 2 2 V(S) = {(a , a ) in C |f(a , a ) = 0 for every f in S}. 1 2 1 2 2 (1) (7 points) Show that T = {C \ V(S)|S: a subset of C[z , z ]} is a 1 2 2 topology on C . (2) (13 points) Are there non-onstant continuous R-valued functions on the 2 topological space (C , T)? Justify your answer. -- 正妹也不過就是一組物質波方程式的特解罷了 --

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