Math 131B Homework 3
Problem 1. Show that the composite of two continuous functions is continuous. Give 3
proofs, using the 3 definitions of continuity.
Problem 2. Suppose d1 and d2 are strongly equivalent metrics on the set X. Prove the
following.
(1) (x
Math 131B Homework 5
Problem 1. Let (X, dX ) and (Y, dY ) be nonempty metric spaces such that Y is complete.
Suppose (fn ) is a sequence of functions fn : (X, dX ) (Y, dY ) which satisfy the criterion:
For every > 0, there is an N > 0 such that m, n > N
Math 131B Homework 4
Problem 1. Let (X,TdX ) be a metric space and
S (E )I be a collection of connected subsets
of X. Suppose that I E 6= . Show that I E is connected.
Hint: Use (continuous) 2-valued functions.
Problem 2. Let (X, dX ) be a metric space, a
Math 131B Homework 2
Problem 1 (Tao 1.3.1). Show that E is closed in the metric space (Y, d|Y Y ) if and only if
E = F Y for some closed F X.
Problem 2. Find a Cauchy sequence in a metric space (X, d) which does not converge. You
must prove that your sequ
Math 131B Homework 1
Problem 1. Tao 1.1.3
Problem 2. Tao 1.1.5
Problem 3. Tao 1.1.8
Problem 4. Tao 1.1.12
Problem 5. Tao 1.2.3
Optional problem 1. Finish the proof that Rn is a metric space under each of the d1 , d2 , d
metrics.
Hint: For the triangle ine