3. Prove or disprove: If a space X is second countable, then every open covering of X contains a countable subcollection covering X.

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1 Department of Mathematics and Statistics University of South Florida TOPOLOGY QUALIFYING EXAM January 24, 2015 Examiners: Dr. M. Elhamdadi, Dr. M. Saito Instructions: For Ph.D. level, complete at least seven problems, at least three problems from each section. For Master s level, complete at least five problems, at least one problem from each section. State all theorems or lemmas you use. Section I: POINT SET TOPOLOGY 1. Show that R ω, countably infinite product of R, with box topology is not metrizable. 2. Let X be a subspace of a compact Hausdorff space Y such that the complement Y \ X of X is a single point. Show that X is locally compact Hausdorff. 3. Prove or disprove: If a space X is second countable, then every open covering of X contains a countable subcollection covering X. 4. Let I = [0, 1] be the unit interval. Prove or disprove: The dictionary order topology on I I is the same as the subspace topology on I I obtained from the dictionary order topology on R R. 5. Let f : R ω R ω be the function given by f((x 1, x 2,...)) = (α 1 x 1 + β 1, α 2 x 2 + β 2,...), where (α 1, α 2,...) and (β 1, β 2,...) are sequences of real numbers with α i > 0 for all i. Show that f is a homeomorphism of R ω with the product topology. What happens if R ω is given the box topology? 6. Let K = { 1 n, n Z+ } and let B be the collection of all open intervals (a, b), along with all sets of the form (a, b) K. The topology generated by B is called the K-topology on R. The space R with this topology is denoted R K. Prove or disprove: R K is a regular space. 7. Prove or disprove each of the following statements: (1) The path components of a space X form a partition of X. (2) Every path component of X is open. 1

2 Section II: ALGEBRAIC TOPOLOGY 1. Prove or disprove: For any positive integer n, there exists a continuous map f : T = S 1 S 1 S 1 from a torus T to a circle S 1 such that the image of the induced homomorphism f on the fundamental groups is a subgroup of index n, i.e., [π 1 (S 1, f(x)) : f (π 1 (T, x))] = n, where x T. 2. Let X be the space obtained by pasting the edges of a polygonal region with the labeling scheme abbca 1 ddc 1. (1) Compute the first homology group H 1 (X). (2) Identify X with one of the surfaces in the list of the Classification Theorem. 3. Compute the fundamental group of the quotient space obtained from the projective plane P 2 by identifying two of its distinct points. 4. Let X be a path connected space such that π 1 (X) is a finite group. Prove that any continuous map f : X S 1 is nullhomotopic. 5. Using a covering map, prove that the fundamental group of the figure-eight (the bouquet of two circles) is not abelian. 6. Let T = S 1 S 1 be a torus. Let A = ({p} S 1 ) (S 1 {q}) be a subspace of T homeomorphic to a figure-eight, where {p} S 1 S 1 S 1 and S 1 {q} S 1 S 1. Determine whether A is a retract of T. 7. Using the -complex, compute the homology groups of the quotient space obtained from the 2-sphere S 2 by identifying three of its distinct points. 2

3 Department of Mathematics and Statistics University of South Florida TOPOLOGY QUALIFYING EXAM September 26, 2015 Examiners: Dr. M. Elhamdadi, Dr. D. Savchuk Instructions: For Ph.D. level, complete at least seven problems, at least three problems from each section. For Master s level, complete at least five problems, at least one problem from each section. State all theorems or lemmas you use. Section I: POINT SET TOPOLOGY 1. Let X be a Hausdorff space and let / X be an additional point. Define a topology on the space Y = X { } by declaring that U Y is open if and only if either U is an open subset of X, or U = Y C, where C is a compact subset of X. Prove that the space Y is a compact space (called the one-point compactification of X). 2. Show that every order topology is regular. 3. Let X be a connected space and assume f : X X is a homeomorphism such that f(f(x)) = x for all x X. Prove that for any continuous function g : X R, there exists x X such that g(x) = g(f(x)). 4. Prove that if X Y then X Int(X) = X Y X. 5. Let X be a topological space and let A be a subset of X. Prove that if there is a sequence of points of A converging to x X, then x A. Prove that the converse holds if X is metrizable. 6. Let f : X Y be a continuous map from the compact metric space (X, d X ) to the metric space (Y, d Y ). Prove that f is uniformly continuous. 7. Show that a connected normal space having more than one point is uncountable. 1

4 Section II: ALGEBRAIC TOPOLOGY 1. Compute the fundamental group of the space obtained by removing four points from the torus S 1 S Prove that any continuous function from the real projective plane P 2 to the circle S 1 is null-homotopic. 3. Let X be the subspace of R 2 that is the union of the three circles of radii one unit centered at (0, 0), (4, 0) and (0, 4) respectively and the line segments from (1, 0) to (3, 0) and the one from (0, 1) to (0, 3). State the Seifert-van Kampen theorem and use it to compute the fundamental group of X. 4. Describe the universal covering of the space S 1 S 2 (wedge of the circle with the 2-sphere). 5. Consider the following surface: X = S 2 #K 2 #K 2 #T 2 #S 2 #P 2 #T 2 #P 2 #T 2 #S 2 (Here P 2 - is a projective plane, K 2 - Klein bottle, T 2 - a torus and S 2 a sphere). (a) Find the presentation for π 1 (X). (b) Find the first homology group of X. (c) Where on the list in the Classification Theorem is X? 6. Compute the simplicial homology groups of the triangular parachute obtained from the 2-simplex 2 by identifying its three vertices to a single point using a simplicial (or -) complex structure and its chain groups. 7. Let A be a deformation retract of X and let x 0 be a point in A. Prove or disprove that the inclusion map j : (A, x 0 ) (X, x 0 ) induces an isomorphism of fundamental groups. 2

5 Department of Mathematics and Statistics University of South Florida TOPOLOGY QUALIFYING EXAM January 23, 2016 Examiners: Dr. M. Elhamdadi, Dr. M. Saito Instructions: For Ph.D. level, complete at least seven problems, at least three problems from each section. For Master s level, complete at least five problems, at least one problem from each section. State all theorems or lemmas you use. Section I: POINT SET TOPOLOGY 1. (1) Define a subbasis S and a basis B for a topology on X, (2) define the topology generated by S, (3) prove that it is indeed a topology, and (4) give an example, with a proof, of a subbasis that generates the standard topology of R 2 and that is not a basis. 2. Let Y be a subspace of X. Prove that a pair of disjoint nonempty sets A and B of Y forms a separation of Y if and only if Y = A B and neither of A nor B contains a limit point of the other. 3. State the definitions of a space being (1) second countable, (2) Lindelöf, and (3) separable. Show that the second countability implies the other two. 4. Let A X; let f : A Y be a continuous map with Y being Hausdorff. Show that if f may be extended to a continuous map g : A Y, then g is uniquely determined by f. 5. Recall that a collection C of subsets of X is said to have the finite intersection property if for every finite subcollection {C 1,..., C n } of C, the intersection C 1 C n is non-empty. State a necessary and sufficient condition, with a proof, of X being compact using the finite intersection property of closed subsets of X. 6. Prove or disprove: Every compact subspace of a normal space is normal. 7. Recall that the square metric on R n is defined by for x = (x 1,..., x n ), y = (y 1,..., y n ) R n. ρ(x, y) = max{ x 1 y 1,..., x n y n } Prove that topologies on R n induced by the square metric and the euclidian metric are the same as the product topology. 1

6 Section II: ALGEBRAIC TOPOLOGY 1. Let p : E B be a covering map. Let e 0, e 1 be points in E such that p(e 0 ) = p(e 1 ) = b 0 B and let H i = p (π 1 (E, e i )) for i = 0, 1. Prove that H 0 and H 1 are conjugate subgroups of π 1 (B, b 0 ). 2. Determine the homology groups of the union of the unit 2-sphere S 2 R 3 and its equatorial disk D = { (x, y, z) z = 0, x 2 + y 2 1 }. 3. Let n > 1 be a positive integer. Provide two spaces X, Y, and a continuous map f : X Y such that the induced map on the first homology groups f : H 1 (X) H 1 (Y ) is an epimorphism Z Z n. 4. Define a covering space and a local homeomorphism. Then provide, with a proof, a local homeomorphism that is not a covering space. 5. Let X be the space obtained by pasting the edges of a polygonal region with the labeling scheme abcdabcd. (1) Compute the first homology group H 1 (X). (2) Identify X with one of the surfaces in the list of the Classification Theorem. 6. Determine the fundamental group of the quotient space obtained from the real projective plane P by identifying three distinct points on P. 7. Prove or disprove: the bouquet of two circles can be embedded in the solid torus S 1 D 2 as a retract. 2

7 Department of Mathematics and Statistics University of South Florida TOPOLOGY QUALIFYING EXAM September 24, 2016 Examiners: Dr. M. Elhamdadi, Dr. M. Saito Instructions: For Ph.D. level, complete at least seven problems, at least three problems from each section. For Master s level, complete at least five problems, at least one problem from each section. State all theorems or lemmas you use. Section I: POINT SET TOPOLOGY 1. Prove or disprove: Let A, B be subspaces of X, Y, respectively. Then the product topology on A B is the same as the subspace topology. 2. Let X = R 2 \ {0} and Y = S 1 ([0, 1] {0}) R 2 be the union of the unit circle and the unit interval on the x-axis. Prove or disprove: X is homeomorphic to Y. 3. Outline a proof that any second countable regular space is metrizable. 4. Let A, B, and A α be subsets of a topological space. Prove or disprove each of the following three equalities: (a) A \ B = A \ B. (b) A α = A α. (c) A α = A α. 5. Let f : X Y be a surjective continuous map. Let X = {f 1 ({y}), y Y } be given the quotient topology. Let g : X Y be the bijective continuous map induced from f (that is, f = g p, where p : X X is the projection map). Prove that the map g : X Y is a homeomorphism if and only if f is a quotient map. 6. Show that any compact Hausdorff space is normal. 7. Let f : S 1 R be a continuous function from the unit circle to the real line. (1) Prove that there exists z 0 S 1, such that f(z 0 ) = f( z 0 ). (2) Determine whether f can be surjective. 1

8 Section II: ALGEBRAIC TOPOLOGY 1. Let L i, i = 1, 2, 3, be three paiwise disjount lines in R 3. Determine the fundamental group of R 3 \ (L 1 L 2 L 3 ). 2. State the equivalence of covering spaces, and classify covering spaces of the circle S Exhibit, with a proof, a space X such that the fundamental group π 1 (X) and the first homology group H 1 (X) are not isomorphic. 4. State the definition of homotpy type of a space. Then prove that the space R m+1 \ {0} and the sphere S m have the same homotopy type. 5. For m 3, prove that R m is not homeomorphic to R For a given non-negative integer n, give the list (with an explanation) of all compact connected surfaces X (with or without boundary) such that the first homology group H 1 (X) has rank n. 7. Determine the homology groups of the following chain complex, where C n = Z, 2n are multiplication by 2 and 2n+1 are zero maps for each n 0: 4 Z 3 =0 Z 2 Z 1=0 Z 0. 2

9 Department of Mathematics and Statistics University of South Florida TOPOLOGY QUALIFYING EXAM May 14, 2016 Examiners: Dr. M. Elhamdadi, Dr. M. Saito Instructions: For Ph.D. level, complete at least seven problems, at least three problems from each section. For Master s level, complete at least five problems, at least one problem from each section. State all theorems or lemmas you use. Section I: POINT SET TOPOLOGY 1. Let K = {1/n n N} R. Let B K = {(a, b), (a, b) \ K a, b R, a < b}. Show that B K forms a basis of a topology, R K, on R, and that R K is strictly finer than the standard topology of R. 2. Recall that a space X is countably compact if any countable open covering of X contains a finite subcovering. Prove that X is countably compact if and only if every nested sequence C 1 C 2 of closed nonempty subsets of X has a nonempty intersection. 3. Assume that all singletons are closed in X. Show that if X is regular, then every pair of points of X has neighborhoods whose closures are disjoint. 4. Let X = R 2 \ {(0, n) n Z} (integer points on the y-axis are removed) and Y = R 2 \ {(0, y) y R} (the y-axis is removed) be subspaces of R 2 with the standard topology. Prove or disprove: There exists a continuous surjection X Y. 5. Let X and Y be metrizable spaces with metrics d X and d Y respectively. Let f : X Y be a function. Prove that f is continuous if and only if given x X and ɛ > 0, there exists δ > 0 such that d X (x, y) < δ d Y (f(x), f(y)) < ɛ. 6. For a surjective map p : X Y, recall that Z X is saturated with respect to p if Z contains every set p 1 ({y}) that it intersects, for any y Y. Assume that p is a quotient map and let Z be a saturated subspace of X with respect to p. Prove that if p is an open map then the map q : Z f(z), obtained by restricting p, is a quotient map. 7. Let f : X Y be a continous function where Y is a Hausdorff space. Prove or disprove the following: The graph of f is a closed subspace of X Y. 1

10 Section II: ALGEBRAIC TOPOLOGY 1. Let S 1 be the complex numbers with unit norm. Let f n : S 1 C \ {0} be defined by f(z) = z n for a positive integer n. Prove that f n is not null-homotopic. 2. Let X be the union of the twelve edges, together with the eight vertices, of the unit cube in 3-space. Determine the fundamental group of X. 3. A space X is said to be contractible if the identity map i X : X X is nulhomotopic. Prove that that a contractible space is path connected. 4. Prove that the n-sphere S n is simply connected for n Let f : X Y, g : Y Z and h : X Z be continuous maps such that h = g f. Prove that if g and h are covering maps then f is also a covering map. 6. Let A, B and C be pairwise distinct points of the plane R 2. Prove or disprove: R 2 \ {A, B, C} is a deformation retract of R 2 \ {A, B}. 7. Let p S 1 and let X be the quotient space of the torus T = S 1 S 1 by the subspace S 1 {p} (where S 1 {p} is identified to a point). Determine all homology groups of the space X. 2