Assignment 6
MAT337
March 2025
1 Rules
1. It is allowed to form teams of at most three members.
2. Each page must contain the full names of all members of team.
3. You must submit your assignment to Crowdmark before 11:00pm of March
31.
4. Justify your arguments in a clear and precise way.
2 Problems
The following result was briefly sketched during lecture. Your first duty is to
show it.
Problem 2.1. Prove that (P(N), d) is a compact space, where d is the metric
defined in the previous assignments.[You already know that the space is com-
plete, so you can try to prove that it is totally bounded. Another alternative
is trying to prove it directly, by modifying the proof for completeness and the
characterization of convergent sequences. ]
Problem 2.2. Consider the subspace X of (C([0, 1]), d∞) given by:
X = {f ∈ C([0, 1]) : ∀x ∈ X (f(x) ∈ [0, 1])}.
Prove that this is space is not compact.
Problem 2.3. Let X and Y be two compact metric spaces. Show that X × Y
is compact [Maybe the easiest way is proving sequential compactness. You will
have to refine a sequence multiple times.]
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