Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2015
Math 636 — Problem Set 3
Issued: 09.18
Due: 09.25
3.1. Let C[0, 1] be the metric space of all continuous functions f : [0, 1] −→
R with the metric d(f, g) = maxx∈[0,1] |f (x) − g(x)|. Is the closed unit
ball {f ∈ C[0, 1] : d(f, 0) ≤ 1}, where 0 is the constant zero function,
compact? Prove.
3.2. Let (X, d) be a compact metric space. Show that it is complete, i.e.,
if a sequence (xn ) is such that for every > 0 there exists n0 such that
d(xn , xm ) < for all n, m ≥ n0 , then xn converges.
3.3. Let (X, d) be a compact metric space. Show that X is separable (i.e.,
has a countable dense subset).
3.4. Let (X, d) be a compact metric space such that for every x, y ∈ X and
every > 0 there exists a sequence x0 = x, x1 , x2 , . . . , xn = y such that
d(xi , xi+1 ) < for all i = 0, 1, . . . , n − 1. Show that X is connected.
(Compare with Problem 2.4.)
3.5. Prove that every countable non-empty Hausdorff compact space has
an isolated point.
3.6. Let S be a well ordered set (i.e., an ordered set such that every subset
has smallest element) with the order topology. Show that if S contains
a maximal element then S is compact.
Order topology on a linearly ordered set S is defined by the basis of
open sets of the form (α, β) = {x ∈ S : α x y}, (−∞, α) = {x ∈
S : x α}, and (α, +∞) = {x ∈ S : x α}.