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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
*x*0 is a point in *A*, then the inclusion of *A*
in *X* induces an
isomorphism pi_1(*A*,*x*0) --> pi_1(*X*,*x*0).
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
(*z*1,*z*2) satisfying |*z*1|^2 + |*z*2|^2 = 1.
Then the circle *S*^1
can be embedded in *S*^3 in two standard ways: *z*1 = 0 is one,
*z*2 = 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.