Problem 1:. (25 points) Suppose that X is a connected T4 space that contains at least
two points. Prove that there is a continuous surjection f : X → [0, 1].
Problem 2:. (25 points) Let X = {0, 1} be the space with two points, equipped with the
discrete topology. Prove that XN is homeomorphic to the Cantor set E from Section I.3,
Exercise 7. (Hint: Think of 0 and 1 as coding left and right.)
Problem 3:. (25 points) Recall that C([0, 1],R) is the metric space of continuous
functions f : [0, 1]→ R equipped with the metric d(f, g) = supt∈[0,1] |f(t)− g(t)|. Prove
that the metric space C([0, 1],R) is path connected.
Problem 4:. (25 points) Recall that B(0; 1) is the closed ball in R2 centered at the origin
of radius 1. Let ∼ be the equivalence relation on B(0; 1) for which p ∼ q for every p and q
with d(0, p) = d(0, q) = 1 (so the points on the circle comprise one equivalence class) while
every element in the interior of the closed ball is only equivalent to itself. Prove that the
quotient space B(0; 1)/ ∼ is homeomorphic to S2 (the unit sphere in R3).
Problem 5: (25 points). Suppose X is a complete metric space and E ⊆ X. Show that E
is totally bounded if and only if E is compact.
Problem 6: (10+15=25 points) Suppose that (X, d) is a metric space and that ρ is a good
metric on X ×X.
(a) Show that the function d : (X ×X, ρ)→ R is continuous.
(b) Suppose that A ⊆ X is compact. Prove that there are a, b ∈ A which are “as far apart
as possible,” that is, d(a′, b′) ≤ d(a, b) for all a′, b′ ∈ A.

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