Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2015
Math 636 — Problem Set 8
Issued: 10.30
Due: 11.06
8.1. Let X be a topological space, and let ∼ be an equivalence relation on
X such that every ∼-equivalence class is dense in X. Show that X/ ∼
has trivial (i.e., anti-discrete) topology.
8.2. Consider the following equivalence relation on R2 . Two points (x1 , y1 )
and (x2 , y2 ) are equivalent if either x1 − x2 and y1 − y2 are integers, or
x1 + x2 and y1 + y2 are integers. Show that the quotient of R2 by this
relation is homeomorphic to the sphere.
8.3. Prove that the group SO(3, R) of orientation preserving orthogonal
transformations of R3 is homeomorphic to the 3-dimensional projective
space. (Hint: every element of SO(3, R) is a rotation by some angle
around some axis.)
8.4. Suppose that {Ui }i∈I is an open cover of a space X such that there
exists a partition of unity subordinate to it. Show that {Ui }i∈I has a
locally finite refinement.
8.5. Let X be a compact Hausdorff space, and let C(X) be the space of continuous functions f : X −→ R with the metric d(f, g) = maxx∈X |f (x)−
g(x)|. Prove that X is second countable if and only if C(X) is separable.