21 January 2015
16.1 Show that if Y is a subspace of X, and A is a subset of Y , then the topology A
inherits as a subspace of Y is the same as the topology it inherits as a subspace
TA,Y = cfw_A U | U TY
11 February 2015
19.7 Let R be the subset of R consisting of all sequences that are eventually
zero, that is, all sequences (x1 , x2 , . . .) such that xi = 0 for only nitely many
values of i. What is the closure of R in R
28 January 2015
17.4 Show that if U is open in X and A is closed in X, then U A is open in X,
and A U is closed in X.
Proof. Since X A is open in X, U A = U (X A) is open in X. Since
X U is closed in X, A U = A (X U ) is cl
6 February 2015
18.2 Suppose that f : X Y is continuous. If x is a limit point of the subset A of
X, is it necessarily true that f (x) is a limit point of f (A)?
Proof. Dene f : R R by
f (x) = 0.
Dene A = (0, 1). Then 0 is
14 January 2015
1.2 (a) A B and A C A (B C).
(b) A B or A C A (B C).
(c) A B and A C A (B C).
(d) A B or A C A (B C).
(e) A (A B) = B.
(f) A (B A) = A B.
4 March 2015
26.2 (a) Show that in the nite complement topology on R, every subspace is
Proof. Let S be a nonempty subspace of R. Let C = cfw_U be a covering
of S by sets open in R. Pick some nonempty U0 C. If S U
18 February 2015
21.7 Let X be a set, and let fn : X R be a sequence of functions. Let be
the uniform metric on the space RX . Show that the sequence (fn ) converges
uniformly to the function f : X R if and only if the sequ
25 February 2015
24.3 Let f : X X be continuous. Show that if X = [0, 1], there is a point x such
that f (x) = x. What happens if X equals [0,1) or (0,1]?
Proof. If f (0) = 0 or f (1) = 1, then f has a xed point. So assume