Course 421: Academic Year 2002–3 Problems I

1. Let X be a non-empty set. Let d: X × X → ℝ be defined so that d(x, y) = 1 if x ≠ y and d(x, y) = 0 if x = y. Verify that the distance function d on X satisfies the metric space axioms.

2. Let X be a metric space with distance function d, and let A be a non-empty subset of X. Let f: X → ℝ be the function defined by f(x) = inf{d(x, a) : a ∈ A} (i.e., f(x) is the largest real number with the property that f(x) ≤ d(x, a) for all a ∈ A). Use the Triangle Inequality to prove that f(x) ≤ f(y) + d(x, y) for all x, y ∈ X, and hence show that |f(x) − f(y)| < d(x, y). (Note that this implies that the function f: X → ℝ is continuous.) Prove that A is closed in X if and only if A = {x ∈ X : f(x) = 0}.

3. Prove that the set

{(x, y) ∈ ℝ^{2} : x ≤ 0 and x^{2} + y^{2} < 1}

is neither open nor closed in ℝ^{2}.

4. Explain why

{(x, y) ∈ ℝ^{2} : y^{3} − 7 < sin(3x^{3} + x^{12} − 57x^{937})}

is an open set in ℝ^{2}.

5. The Zariski topology on the real numbers ℝ is the topology whose open sets are the empty set, the set ℝ itself and those subsets of ℝ whose complements are finite.

(a) Verify that the Zariski topology is a well-defined topology on the set ℝ of real numbers (i.e., show that the topological space axioms are satisfied).

(b) Prove that any polynomial function from ℝ to itself is continuous with respect to the Zariski topology in ℝ.

(c) Give an example of a function from ℝ to itself which is continuous with respect to the usual topology on ℝ but is not continuous with respect to the Zariski topology on ℝ.

6. (a) Let f: X → Y be a function from a topological space X to a topological space Y, and let A and B be subsets of X for which X = A∪B. Suppose that the restrictions f|A and f|B of f to the sets A and B are continuous. Is f: X → Y necessarily continuous on X? [Give proof or counterexample.]

(b) Let f: X → Y be a function from a topological space X to a topological space Y, and let F be a (not necessarily finite) collection of closed subsets of X whose union is the whole of X. Suppose that the restriction f|A of f to A is continuous for all closed sets A in the collection F. Is f: X → Y necessarily continuous on X? [Give proof or counterexample.]

7. Let f: X → Y be a function from a topological space X to a topological space Y, and let U be a collection of open subsets of X whose union is the whole of X. Suppose that the restriction f|W of f to W is continuous for all open sets W in the collection U. Prove that f: X → Y is continuous on X.

8. Let X be a topological space, let A be a subset of X, and let B be the complement X \ A of A in X. Prove that the interior of B is the complement of the closure of A.

9. Determine which of the following subsets of ℝ^{3} are compact.

(i) The x-axis {(x, y, z) ∈ ℝ^{3} : y = z = 0}

(ii) The surface of a tetrahedron in ℝ^{3}.

(iii) {(x, y, z) ∈ ℝ^{3} : x > 0 and x^{2} + y^{2} − z^{2} ≤ 1}.

10. Let X be a topological space. Suppose that X = A ∪ B, where A and B are path-connected subsets of X and A∩B is non-empty. Show that X is path-connected.

11. Let f: X → Y be a continuous map between topological spaces X and Y. Suppose that X is path-connected. Prove that the image f(X) of the map f is also path-connected.

12. Let X and Y be path-connected topological spaces. Explain why the Cartesian product X × Y of X and Y is path-connected.

13. Determine the connected components of the following subsets of ℝ^{2}:

(i) {(x, y) ∈ ℝ^{2} : x^{2} + y^{2} = 1}, (ii) {(x, y) ∈ ℝ^{2} : x^{2} − y^{2} = 1}, (iii) {(x, y) ∈ ℝ^{2} : y^{2} = x(x^{2} − 1)}, (iv) \{(x, y) ∈ ℝ^{2} : (x − n)^{2} + y^{2} > \frac{1}{4} for all n ∈ ℤ\}. [Fully justify your answers.]

14. A topological space X is said to be locally path-connected if, given any point x of X there exists a path-connected open set U in X such that x ∈ U.

(a) Let X be a locally path-connected topological space, and let p be a point of X. Let A be the set of all points x of X for which there exists a path from p to x, and let B be the complement of A in X. Prove that A and B are open in X.

(b) Use the result of (a) to show that any connected and locally pathconnected topological space is path-connected.

15. Let X be a convex subset of ℝ^{n}. (A subset X of ℝ^{n} is said to be convex if (1 − t)x + ty ∈ X for all x, y ∈ X and real numbers t satisfying 0 ≤ t ≤ 1.)

(a) Prove that any two continuous functions mapping some topological space into X are homotopic.

(b) Prove that any two continuous functions mapping X into some path-connected topological space Y are homotopic.

16. Determine which of the following maps are covering maps:— (i) the map from ℝ to [−1, 1] sending θ to sin θ, (ii) the map from S^{1} to S^{1} sending (cos θ, sin θ) to (cos nθ, sin nθ), where n is some non-zero integer, (iii) the map from {z ∈ ℂ : Re z < 0} to {z ∈ ℂ : 0 < |z| < 1} sending z to exp(z), (iv) the map from {z ∈ ℂ : −4π < Im z < 4π} to {z ∈ ℂ : |z| > 0} sending z to exp(z).

[Briefly justify your answers.]

17. A continuous function f: X → Y between topological spaces X and Y is said to be a local homeomorphism if, given any point x of X there exists an open set V in X containing the point x and an open set W in Y containing the point f(x) such that the function f maps V homeomorphically onto W. Explain why any covering map is a local homeomorphism.

18. Determine which of the maps described in question 16 are local homeomorphisms.

19. (a) Let W, X, Y and Z be topological spaces, and let A be a subset of X. Let f: X → Y and g: X → Y be continuous maps. Suppose that f ≃ g rel A. Show that h ∘ f ≃ h ∘ g rel A for all continuous maps h: Y → Z, and that f ∘ e ≃ g ∘ e rel e^{−1}(A) for all continuous maps e: W → X.

(b) Using (a), explain why, given any continuous map f: X → Y between topological spaces X and Y, there is a well-defined homomorphism f_{#}: π_{1}(X, x) → π_{1}(Y, f(x)) of fundamental groups for any x ∈ X which sends [γ] to [f ∘ γ] for any loop γ based at the point x.

(c) Let f: X → Y and g: X → Y be continuous maps satisfying f(x) = g(x) and f ≃ g rel{x}. Show that the homomorphisms f_{#} and g_{#} from π_{1}(X, x) to π_{1}(Y, f(x)) induced by the maps f and g are equal.

20. (a) Let X and Y be topological spaces, let f: X → Y and h: Y → X continuous maps, and let x be a point of X. Suppose that h(f(x)) = x and that h ∘ f ≃ 1_{X} rel{x} and f ∘ h ≃ 1_{Y} rel{f(x)}, where 1_{X} and 1_{Y} denote the identity maps of the spaces X and Y. Explain why the fundamental groups π_{1}(X, x) and π_{1}(Y, f(x)) are isomorphic.

(b) Using (a), explain why the fundamental groups π_{1}(ℝ^{n}\{0}, p) and π^{1}(S^{n−1}, p) of ℝ^{n} \ {0} and the (n − 1)-dimensional sphere S^{n−1} are isomorphic for all n > 1, where p ∈ S^{n−1}.

21. Let X be a topological space, and let α: [0, 1] → X and β: [0, 1] → X be paths in X. We say that the path β is a reparameterization of the path α if there exists a strictly increasing continuous function σ: [0, 1] → [0, 1] such that σ(0) = 0, σ(1) = 1 and β = α ∘ σ. (Note that if β is a reparameterization of α then α(0) = β(0), α(1) = β(1), and the paths α and β have the same image in X.)

(a) Show that there is a well-defined equivalence relation on the set of all paths in X, where a path α is related to a path β if and only if β is a reparameterization of a path α. [Hint: use the basic result of analysis which states that a strictly increasing continuous function mapping one interval onto another has a continuous inverse.]

(b) Show that if the path β is a reparameterization of the path α, then β ≃ α rel{0, 1}.

Given paths γ_{1}, γ_{2}, …, γ_{n} in a topological space X, where γ_{i}(1) = γ_{i+1}(0) for i = 1, 2, …, n−1, we define the concatenation γ_{1}.γ_{2}. · · · .γ_{n} of the paths by the formula (γ_{1}.γ_{2}. . . . .γ_{n})(t) = γ_{i}(nt − i + 1) for all t satisfying (i − 1)/n ≤ t ≤ i/n.

(c) Show that the path (γ_{1}. . . . .γ_{r}).(γ_{r+1}. . . . .γ_{n}) is a reparameterization of γ_{1}.γ_{2}. · · · .γ_{n} for any r between 1 and n − 1.

(d) By making repeated applications of (c), or otherwise, show that (γ_{1}.γ_{2}).γ_{3}.(γ_{4}.γ_{5}) is a reparameterization of γ_{1}.(γ_{2}.γ_{3}.γ_{4}).γ_{5} for all paths γ_{1}, γ_{2}, γ_{3}, γ_{4}, γ_{5} in X satisfying γ_{i}(1) = γ_{i+1}(0) for i = 1, 2, 3, 4.

22. Let X be a topological space.

(a) Show that, given any path α: [0, 1] → X in X, there is a welldefined homomorphism Θ_{α}: π_{1}(X, α(1)) → π_{1}(X, α(0)) of fundamental groups which sends the homotopy class [γ] of any loop γ based at α(1) to the homotopy class [α.γ.α^{−1}] of the loop α.γ.α^{−1}, where

(α.γ.α^{−1})(t) = \{ \begin{matrix}α(3t) & if 0 ≤ t ≤ \frac{1}{3}, \\ γ(3t − 1) & if \frac{1}{3} ≤ t ≤ \frac{2}{3}, \\ α(3 − 3t) & if \frac{2}{3} ≤ t ≤ 1 \end{matrix}

(i.e., α.γ.α^{−1} represents ‘α followed by γ followed by α reversed’).

(b) Show that Θ_{α.β} = Θ_{α} ∘ Θ_{β} for all paths α and β in X satisfying β(0) = α(1).

(c) Show that Θ_{α} is the identity homomorphism whenever α is a constant path.

(d) Let α and α̂ be paths in X satisfying α(0) = α̂(0) and α(1) = α̂(1). Suppose that α(0) ≃ α̂(0) rel{0, 1}. Show that Θ_{α} = Θα̂.

(e) Explain why the homomorphism Θ_{α}: π_{1}(X, α(1)) → π_{1}(X, α(0)) is an isomorphism for all paths α in X. (This shows that, up to isomorphism, the fundamental group of a path-connected topological space does not depend on the choice of basepoint.)

23. Let X and Y be topological spaces, and let x_{0} and y_{0} be points of X and Y. Prove that π_{1}(X × Y, (x_{0}, y_{0})) ≅ π_{1}(X, x_{0}) × π_{1}(Y, y_{0}). [Hint: you should make use of the result that a function mapping a topological space into X × Y is continuous if and only if its components are continuous.]

24. (a) Let p: X̃ → X be a covering map, let α: [0, 1] → X and β: [0, 1] → X be paths in X, and let α̃: [0, 1] → X̃ and β̃: [0, 1] → X̃ be lifts of α and β satisfying p ∘ α̃ = α, p ∘ β̃ = β and α̃(0) = β̃(0). Suppose that α ≃ β rel{0, 1}. Prove that α̃(1) = β̃(1).

(b) Let p: X̃ → X be a covering map. Suppose that X̃ is pathconnected and X is simply-connected. Using the results of (a), or otherwise, prove that p: X̃ → X is a bijection, and is thus a homeomorphism.

25. Real projective ℝP^{n} n-dimensional space may be described as the topological space obtained from the n-dimensional sphere S^{n} by identifying together each pair of antipodal points on S^{n}. There is thus an identification map q: S^{n} → ℝP^{n} that sends each pair of antipodal points of S^{n} to the corresponding point of ℝ^{n}. One can easily verify that the map q is a covering map.

Prove that the fundamental group of ℝP^{n} is isomorphic to ℤ_{2} when n ≥ 2, where ℤ_{2} is the group with exactly two elements. [Hint: adapt the proof of the theorem concerning the fundamental group of the circle.]