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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.