Math 363 - Topics in Topology
Assignment 2 – September 10, 2021
Subspace Topology, Connectedness, and Continuity
Due Friday, September 17, 2021.
1. Subspace topologies. Consider (R2,U). Let
Y = {(x, y) : |x| − 1 < y ≤ 1− |x| and (x, y) 6= (0, 0) }
with the subspace topology UY . For each of the following subsets S of Y , determine
the interior, closure, and boundary of S in (Y,UY )
(a) S = {(x, y) : 0 < y < 1− |x| }.
(b) S = {(x, y) : 0 < x < 1− |y| }.
(c) S = {(x, y) : xy = 0, |x| < 1, |y| < 1, (x, y) 6= (0, 0) }
2. Subspace topologies. Let S be the subset of R2 given by
S = {(x, y) : x2 + y2 < 1, x ≥ 0, and y > 0}.
For each of the following sets Y ⊂ R2 containing S, determine whether S is an open
set, closed set, or neither as a subset of Y considered as a topological space with the
subset topology from (Y,UY ). Give a
ief, descriptive explanation of your answer.
(a) Y = {(x, y) : y > 0}.
(b) Y = {(x, y) : x ≥ 0}.
(c) Y = {(x, y) : x2 + y2 < 1 and y > 0}.
3. Connectedness. Which of the following subsets of (R2,U) are connected? If the set
is connected,
iefly explain why. If not, give two open sets of R2 that separate the
set.
(a) The Target symbol, S = {(x, y) : 1 ≤ x2 + y2 ≤ 4 or 9 ≤ x2 + y2 ≤ 16}.
(b) S = {(x, y) : x2 + y2 = 1, y > 0} ∪ {(x, y) : |x| < 1 and 0 ≥ y ≥ −1}.
(c) S = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and x, y ∈ Q}.
4. Connectedness. Either prove the following statements or give a counterexample.
Assume that A and B are not empty. (You may work in (R2,U) if you wish.)
(a) If A and B are connected then A ∩B is connected.
(b) If A and B are disconnected, the A ∩B is disconnected.
(c) If A is connected, then A is connected.
(d) If A is connected, then Int(A) is connected.
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5. Composition of continuous functions. Suppose that (X,S), (Y, T ), and (Z,U) are
topological spaces. Further, suppose that f : X → Y and g : Y → Z are continuous
functions. Prove that the composition g ◦ f is continuous. (Hint: All you have to work
with is the topological definition of continuity.)
6. Continuity and connectedness. Suppose that (X,S) and (Y, T ) are topological
spaces and that f : X → Y is continuous. For a subset A ⊂ X, the image of A unde
f , denoted f(A), is defined by:
f(A) = {y ∈ Y : ∃x ∈ A, f(x) = y }.
(a) Prove that if f(A) ⊂ Y is disconnected, then A is also disconnected.
(b) Use (a) to prove that if A is connected, the f(A) is also connected.
(c) Assume now that f is a homeomorphism. Prove that A is connected if and only
if f(A) is connected.
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