Download Assignment 4.

MATHEMATICS 365 Spring 2017
Assignment #4
Due: Please hand in your own solutions at the start of class on
Friday, March 3. Late assignments will not be accepted.
1. Let G be a group equipped with a Hausdorff topology in which the group
operations G × G → G, (g, h) 7→ gh, and inversion G → G, g 7→ g −1 are
continuous. This is what is called a topological group. Let H be a subgroup.
We define an equivalence relation on G by g1 ∼ g2 if gh1 = g2 for some h ∈ H,
and let G/H denote the set of equivalence classes, as is usual in group theory.
On G/H we put the quotient topology.
a) Prove that if H is open in G (as a subset) then G/H has the discrete
topology.
b) Prove that if H is dense in G then G/H has the trivial topology.
c) Prove that a subset A ⊂ G is saturated if and only if it is invariant under
right translation by H, i.e. if and only if AH = A, where AH = {ah | a ∈
A, h ∈ H}.
d) Give the group R of real numbers its standard topology (it is the a topological group). The group Z of integers is a subgroup. Prove that R/Z with
its quotient topology is homeomorphic to the circle T with its standard
topology.
2. Let ∼ be the equivalence relation on [0, 1] × {0, 1} defined (x, 0) ∼ (x, 1) if
x > 0, and x ∼ x for all x. Let X ∗ be the corresponding quotient space with
the quotient topology. Prove that X ∗ is not Hausdorff.
3. Let X = [0, 1] with its standard topology, and let Y = [0, 1] × [−1, 1] with its
standard topology. Consider the equivalence relation on X which declares 0
and 1 equivalent (and sets every point equivalent to itself.) Let X ∗ be the corresponding quotient space. On Y we define an equivalence relation by declaring
(0, t) ∼ (1, −t) for all t ∈ [−1, 1], and every point equivalent to itself; let Y ∗ be
the set of equivalence classes.
a) Argue by suitable pictures (not a high level of precision is needed here)
that X ∗ is homeomorphic to the circle, and that Y ∗ is homeomorphic to
the Möbius band.
b) Let π : Y ∗ → X ∗ be the map π([(x, t)]) = [x]. Prove that π is a quotient
map.
c) Prove using the Intermediate Value Theorem that if f : X ∗ → Y ∗ is a
continuous map such that π ◦ f = π, then the image of f has non-empty
intersection with the slice {[(x, 0)] |; x ∈ [0, 1]} ⊂ Y ∗ .