## Homework

Make sure you can do all the exercises in those parts of the text that we cover.

Due Sept. 7 H&Y exercises 1-6,9,10,12

Due Sept. 14 H&Y exercises 1-14,16,17.
Prove that a continuous function defined on a closed, finite interval takes on maximum and minimum values.
Prove the "intermediate value theorem:" if f is a continuous function defined on the closed interval [a,b], and c is a real number between f(a) and f(b), then there exists x in the interval such that f(x) = c.
Prove that the subset topology on the subset A of the space X is the coarsest topology on A for which the inclusion map i:A-->X (defined by i(x) = x for x in A) is continuous.

Due Sept. 21 H&Y exercises 1-18,19,20; Kelley Problem D p. 162; H&Y 2-1,2,3,8.

Due Sept. 28 For i=0,...,4 give an example of a topological space that is T(i) and not T(i+1). You may use references and work together, but you must give exact citations and tell the grader who your partners are.

Due Oct. 5 Two problems on topological groups (see H&Y, p. 33 for definition)
1. (Kelley Problem T p. 106, a...e) Prove:
a. A subgroup of a topological group is a topological group (in the subspace topology).
b. The closure of a subgroup is a subgroup and the closure of a *normal subgroup is *normal. (*normal in the algebraic sense: xN = Nx for x in G)
c. Every subgroup with non-empty interior is open and closed. A subgroup H either is closed or satisfies: the closure of H, minus H, is dense in the closure of H.
d. The smallest subgroup which contains a fixed open subset of a topological group is both open and closed.
e. The connected component of the identity in a topological group is a *normal subgroup.
2. Prove that every Hausdorff topological group is completely regular. (Munkres p.237, ex. 5. Follow his hints!)

Due Oct. 12
1. a. Prove that the torus T^2 is an ANR. (Give an elementary proof; do not appeal to the discussion on normal bundles etc.) b. Generalize from T^1 = S^1 to the n-torus T^n.
2. a. Prove that any finite set of points is an ANR. b. Prove that the set {0} union {1/n, n = 1,...,\infty} is not a neighborhood retract in [-1,2].
3. Write a paragraph about the role of the real numbers, and the l.u.b. axiom in particular, in the theory of retracts.

Due Oct. 24
1. Prove that a continuous, injective map f: X --> Y, X compact, Y Hausforff, is a homeomorphism from X to f(X).
1'. Be familiar with examples of continuous, injective maps which are not homeomorphisms onto their images!
2,3,4. Three problems on paracompactness and partitions of unity from Munkres; p. 225, Exercises 3a,4,5.

Due Oct. 26
Construct simple coverings for R^2, S^1, S^2, S^3, T^2, T^3. Try to construct simple coverings with the smallest possible number of elements. When you can, prove that you have achieved this goal.

Due Nov. 2
Prove the existence of "Lebesgue numbers." I.e. given a covering by open sets of a compact metric space X, there exists a positive number r such that for any point x in X the ball B_r(x) of radius r about x is completely contained in one of the open sets of the covering.

Due Nov. 16
H & Y 4.2. Check the calculations on pp 161,163,164-5.
In the following three problems, answer by giving explicit homotopies.
Let S^1 = {|z|=1} in the complex plane C, and let f: S^1 --> C-{0} be given by f(z) = 2+z. Show f is homotopic to a constant.
Let S^1 be parametrized by the angle theta, 0 < = theta < = 2 pi, and let (phi, theta) be the usual (co-latitude, longitude) spherical coordinates on S^2. Show that the map f: S^1 --> S^2 given by f(theta) = (pi/2, theta) is homotopic to a constant.
Let S^3 be the set {x^2+y^2+z^2+w^2 = 1} in R^4, and let f:S^3 --> S^3 be given by f(x,y,z,w) = (-x,-y,-z,-w) (the antipodal map). Show that f is homotopic to the identity map.

Due Nov. 21
1. Prove that if A is contained in X as a deformation retract, and x0 is a point in A, then the inclusion of A in X induces an isomorphism pi_1(A,x0) --> pi_1(X,x0). Assuming known that pi_1(circle) = Z, calculate pi_1(plane minus a point) and pi_1(solid torus).
2. The standard torus in R^3 is the boundary of a ring (the solid torus). Assuming known that pi_1(torus) = Z + Z, prove that there cannot be a retraction of the solid torus onto its boundary.
3. Take the 3-sphere S^3 as the set of pairs of complex numbers (z1,z2) satisfying |z1|^2 + |z2|^2 = 1. Then the circle S^1 can be embedded in S^3 in two standard ways: z1 = 0 is one, z2 = 0 is the other. Show that S^3 minus either one has the other as deformation retract. In particular, use this to calculate pi_1(S^3 minus an unknotted circle).
4. Library research. What is pi_1(S^3 minus a trefoil knot)?

Due Nov. 30
Seven problems from Munkres.
p. 336 nos. 1,2,4,5,6
p. 341 nos. 8,9.

Due Dec. 7
Munkres p. 398 nos 1-6.