Josiah Oh
Topology
Assignment 2
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
Josiah Oh
Topology
Assignment 5
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
Josiah Oh
Topology
Assignment 3
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 op
Josiah Oh
Topology
Assignment 4
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.
Josiah Oh
Topology
Assignment 1
14 January 2015
1.2 (a) A B and A C A (B C).
False. holds.
(b) A B or A C A (B C).
False. holds.
(c) A B and A C A (B C).
True.
(d) A B or A C A (B C).
False. holds.
(e
Josiah Oh
Topology
Assignment 8
4 March 2015
26.2 (a) Show that in the nite complement topology on R, every subspace is
compact.
Proof. Let S be a nonempty subspace of R. Let C = cfw_U be a covering
Josiah Oh
Topology
Assignment 6
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
un
Josiah Oh
Topology
Assignment 7
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 (