Homework 6
Math 501
Due October 10, 2014
Exercise 1
Let M be a metric space and let N = {0, 1} be a metric space by giving it
the metric inherited from R. Prove that M is connected if and only if every
continuous function f : M → N is constant.
Exercise 2
We proved in class that if N ⊂ M , then U ⊂ N is open in N if and only
there exists a set V ⊂ M open in M such that U = V ∩ N . A simple argument
taking complements shows that K ⊂ N is closed in N if and only if there exists
a set L ⊂ M closed in M such that K = L ∩ N .
Use the above to show that
(a) If N is closed in M , then K ⊂ N is closed in N if and only if K is closed in
M.
(b) If N is open in M , then U ⊂ N is open in N if and only if U is open in M .
Exercise 3
Let f : M → R be a function. The graph of f is the set Γ = {(p, f (p)) ∈
M × R : p ∈ M }.
(a) Prove that if f is continuous then its graph is closed in M × R.
(b) Show that the set H = {(x, y) ∈ R2 : xy = 1, x, y > 0} is closed in R2 .
(c) Let X = {(x, 0) ∈ R2 : x ∈ R}. Argue that H ∪ X is closed in R2 .
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(d) Let S = H ∪ X. Show that H and X are both closed in S. Conclude that
S is disconnected.
Remark. The above argument generalizes to show that if Γ is the graph of any
continuous function f : R → (0, ∞), then Γ ∪ X is disconnected. This is not
very surprising. However, if f is discontinuous at just a single point, this fact
no longer holds. For example, consider the function
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x≤0
f (x) =
(x + 1) sin x1 x > 0.
For this function Γ ∪ X is in fact connected. The argument for this is similar to
the argument which shows that the topologist’s sine curve is connected.
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