Download Topology, MM8002/SF2721, Spring 2017. Exercise set 4 Exercise 1

Topology, MM8002/SF2721, Spring 2017. Exercise set 4
Exercise 1. Let X be topological space, Y be a set and f : X → Y be a surjective
map. Recall that a subset U ⊆ Y is open in the quotient topology, if and only if
f −1 (U ) is open in X.
• Show that the quotient topology is in fact a topology.
• Show that the quotient topology is the finest topology making f continuous.
Exercise 2. Let f : X → Y be a continuous surjective function.
• Let f be open. Show that it is a quotient map.
• Let f be closed. Show that it is a quotient map.
Exercise 3. Let G be a group and a topological space.
• Show that G is a topological group if and only if
G × G → G, (x, y) 7→ xy −1
is continuous.
Exercise 4. Consider Z ⊆ R as a subset.
• Show that the quotient by collapsing Z to a point gives a space homeomorphic to a countable wedge of circles.
• Show that the quotient is not first countable.
Exercise 5. Consider Z ⊆ R as a subgroup.
• Show that the quotient R/Z is homeomorphic to the circle.
Exercise 6. The orthogonal group O(n) acts on Rn by matrix multiplication.
• Show that the quotient is homeomorphic to the non-negative real numbers.
Exercise 7. Consider the Möbius strip
[−1, 1] × [−1, 1]/ ∼ , where (−1, x) ∼ (1, −x).
We can define an inclusion of the circle
ι : [−1, 1]/{−1, 1} → [−1, 1] × [−1, 1]/ ∼,
induced by the inclusion
[−1, 1] → [−1, 1] × [−1, 1], x 7→ (x, 0).
• Show that ι is well defined and an embedding.
We can also define a ‘projection’ onto the circle
π : [−1, 1] × [−1, 1]/ ∼ → [−1, 1]/{−1, 1},
induced by
[−1, 1] × [−1, 1] → [−1, 1], (x, y) 7→ x.
• Show that π well defined and a quotient map.
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