Quiz 1
Answer EXACTLY one of the following four questions. Each question is worth 20 points. The
remaining three questions are to be submitted as homework. Please DO NOT attempt more
than one question in the quiz; if you do, there will be a penalty of five points.
1. Let ω := e
2πi
4
. Define the set Y to be
Y := S 1 / ∼
where
z ∼ ωz.
Give Y the quotient topology induced from S 1 . Show that Y is homeomorphic to S 1 .
2. Define the set Y to be
Y := R2 / ∼
where
(x, y) ∼ (x + 1, y).
2
Give Y the quotient topology induced from R . Show that Y is homeomorphic to S 1 × R.
3. Consider the following map f : [0, 1) −→ S 1 given by
f (x) := e2πix .
Notice that f is a bijective continuous map. Show that f −1 is not continuous (hence f is not a homeomorphism).
4. Define the set Y to be
Y := [0, 1] × [0, 1]/ ∼
where
(0, y) ∼ (1, y)
and
(x, 0) ∼ (x, 1).
Give Y the quotient topology induced from R2 . Show that Y is homeomorphic to S 1 × S 1 .
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