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MATH 358 – FINAL EXAM REVIEW The following is designed to help you prepare for the final exam. A substantial portion of the final will be derived from these questions and their close relatives. 1. What does it mean? Give definitions of each of the following. Where appropriate, provide a short, illustrative example. • The preimage of a set U under function f . . . • The image of function f . . . • The graph of a function f : X → Y . . . • A set X is countable. . . • A relation on X is an order relation. . . • A topology on set X. . . • A basis on set X. . . • The topology generated by a basis. . . • The order topology. . . • The product topology. . . • The subspace topology. . . • The quotient topology. . . • Topology T1 on set X is finer than topology T2 . . . • A function f : X → Y is continuous. . . • A subset A of topological space X is closed . . . • Let A be a subset of X. The interior int(A). . . • Let A be a subset of X. The closure A. . . • The standard topology on R. . . • The lower-limit topology on R. . . • The finite-complement topology on R. . . • The dictionary order on X × Y . . . • The dictionary order on Z2 . . . • A function f : X → Y is a homeomorphism. . . • A topological space X is connected . . . • A topological space X is path-connected . . . • A topological space X is Hausdorff . . . • A topological space X is compact. . . • A subset K of topological space X is compact. . . • A homotopy between functions f and g. . . • A path homotopy. . . • A set K in Rn is convex . . . • A topological space X is simply connected . . . Date: Fall 2013 . 1 2 MATH 358 – FINAL EXAM REVIEW • The fundamental group π1 (X) of a topological space X. . . • A lift of f : [a, b] → S 1 . . . • The winding number of f : [a, b] → S 1 . . . • A retract A of X. . . 2. True or False? Determine if each of the following statements are true of false. If a statement is true, give a short sketch of the proof; if false, provide a counter-example. All sets X, Y , etc. are assumed to be topological spaces unless otherwise specified. (1) The set of rationals is countable. (2) The set {3, 4, 5}ω is countable. (3) Countable unions of countable sets are countable. (4) The relation R on Z+ given by aRb iff a2 < b is an order relation. (5) If each Uα are open in topological space X for α in index set J, then ∪α Uα is open in X. (6) If each Uα are closed in topological space X for α in index set J, then ∪α Uα is closed in X. (7) If each Uα are open in topological space X for α in index set J, then ∩α Uα is open in X. (8) If each Uα are closed in topological space X for α in index set J, then ∩α Uα is closed in X. (9) There exists a largest topology on set X. (10) There exists a smallest topology on set X. (11) Let X be a topological space and give Y ⊂ X the subset topology. If U ⊂ Y is open in Y , then it is also open in X. (12) (13) {(a, b) × (c, d) | a, b, c, d ∈ R} is a basis for the standard topology on R2 . (14) For x ∈ R2 and r ∈ (0, ∞), define Br (x) = {y ∈ R2 | kx − yk < r}. The collection B = {Br (x)} forms a basis for the standard topology on R2 . (15) The collection {U ⊂ X | x \ U is empty or infinite or all of X} is a topology on X. (16) If Tz is a topology on X for each z ∈ R, then ∩z Tz is also a topology on X. (17) If Tz is a topology on X for each z ∈ R, then ∪z Tz is also a topology on X. (18) The set U = ∪k∈Z (k − 71 , k + 17 ) is open in Rl . (19) (1, 2) × (0, 1) ∪ (3, 4) × (0, 1) is an element of the basis for the product topology on R × R. (20) All functions are continuous. (21) If f : X → Y is continuous, then f (X) is open. (22) If f : X → Y is continuous, then f (X) is not open. (23) If f : X → Y is continuous, then f (X) is closed. (24) If f : X → Y is continuous, then f −1 (Y ) is open. (25) If f : X → Y is continuous, then f −1 (Y ) is not open. (26) If f : X → Y is continuous, then f −1 (Y ) is closed. (27) If the graph of f : X → Y is closed, then f is continuous. (28) If f : X → Y is continuous, then the graph of f is closed. (29) If f : X → Y is a continuous bijection, then X and Y are homeomorphic. (30) Homeomorphism is an equivalence relation on the set of topological spaces. (31) Constant functions are continuous. (32) If A ⊂ X has the subspace topology then the embedding j : A → X is continuous. MATH 358 – FINAL EXAM REVIEW 3 (33) The functions f : X → Y and g : Y → Z are continuous if and only if the composite f ◦ g : X → Z is continuous. (34) The integers are connected with respect to the topology induced by Rstd . (35) The rationals are connected with respect to the topology induced by Rstd . (36) The graph of a function is connected (with respect to the subspace topology) if and only if the function is continuous. (37) X × Y is connected if and only if X and Y are connected. (38) The intervals are the only bounded connected subsets of Rstd . (39) If a space is path-connected, then it is connected. (40) If a space is connected, then it is path-connected. (41) If X is Hausdorff and A ⊂ X is finite, then A is closed. (42) If X is Hausdorff then X is connected. (43) Closed subspaces of compact spaces are compact. (44) If K is a compact subset of Hausdorff space X and y ∈ X \ K then there exists open U, V with K ⊂ U , y ∈ V , and U ∩ V = ∅. (45) Any topological space contains at least one compact subset. (46) Any interval of Rstd contains a non-compact subset. (47) If Y ⊂ X has the subspace topology and K is a compact subset of Y , then K is also a compact subset of X. (48) If Kj are compact subsets of Rstd for j ∈ Z+ , then ∪j∈Z+ Kk is also compact. (49) If Kj are compact subsets of Rstd for j ∈ Z+ , then ∩j∈Z+ Kk is also compact. (50) If K ⊂ R is compact with respect to the standard topology, then it is also compact with respect to the lower-limit topology. (51) If f : X → Y is continuous and X is connected, then so is f (X). (52) If f : X → Y is continuous and Y is connected, then so is f −1 (Y ). (53) If f : X → Y is continuous and X is path-connected, then so is f (X). (54) If f : X → Y is continuous and Y is path-connected, then so is f −1 (Y ). (55) If f : X → Y is continuous and X is compact, then so is f (X). (56) If f : X → Y is continuous and Y is compact, then so is f −1 (Y ). (57) Let f : X → Y be a continuous bijection. Then X is compact if and only if Y is. (58) Path homotopy is an equivalence relation on the collection of all paths from point P to point Q. (59) If f : K → X is continuous and K ⊂ Rn is convex, then f is homotopy equivalent to a constant function. (60) If f : K → X is continuous and K ⊂ Rn is compact, then f is homotopy equivalent to a constant function. (61) All convex sets are simply connected. (62) All simply connected sets are convex. (63) If X \ {∗} is simply connected, then so is X. (64) Suppose f : [0, 1] → S 1 has a winding number of zero. Then f is not surjective. (65) Suppose f : [0, 1] → S 1 is injective. Then the winding number of f is zero. (66) Let n ≥ 0 and suppose f : [0, 1] → S n is not surjective. Then f is nullhomotopic. (67) S 1 is a retract of R2 . (68) S 1 is a retract of R2 \ {∗}. 4 MATH 358 – FINAL EXAM REVIEW (69) S 1 is a retract of T 2 . (70) S 1 is a retract of T 2 \ {∗}. (71) S 1 is a retract of S 2 . (72) S 1 is a retract of S 2 \ {∗}. (73) A space X is simply connected if and only if S 1 is a retract of X. (74) Connectedness is a homeomorphism invariant. (75) Path-connectedness is a homeomorphism invariant. (76) Compactness is a homeomorphism invariant. (77) Simple connectedness is a homeomorphism invariant. (78) Suppose X and Y are compact and simply connected. Then X ∼ =Y. 3. Examples I encourage you to make a long list of examples, and types of examples, we have encountered. You should be able to: • Give the basic properties of each space (compact, connected, etc.) • Give the basic properties of subsets of each space. • Determine which spaces are homeomorphic and, if they are, present (sketches of) homeomorphisms. • Determine which spaces are not homeomorphic and identify a distinguishing characteristic or property.