Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2015
Math 636 — Problem Set 5
Issued: 10.09
Due: 10.16
5.1. Show that ex is not uniformly continuous with respect to the usual
uniform structure on R.
5.2. Let X a discrete space, and let X D be the set of all maps D −→ X.
For a finite subset A ⊂ I of the set of indices, denote by VA the relation
on X D given by the rule:
(f, g) ∈ VA ⇐⇒ f |A = g|A .
Show that the set of all such forms a basis of entourages for a uniform
structure on X D . Prove that the topology associated with it coincides
with the direct product topology.
5.3. Show that intersection of all entourages of a uniform structure is an
equivalence relation.
5.4. Let `∞ be the space of all bounded sequences (x1 , x2 , . . .) of real numbers with the metric
d((xn ), (yn )) = sup |xn − yn |.
Is the set {(xn ) : |xn | ≤ 2−n } compact? You can use the fact that `∞
is complete.
5.5. Let G be the group of affine maps x 7→ ax + b for real numbers
a 6= 0 and b. For a neighborhood U of (1, 0) in R2 consider the sets RU
(respectively LU ) of pairs (g1 , g2 ) ∈ G×G such that g1 g2−1 (respectively
g1−1 g2 ) is equal to ax + b for (a, b) ∈ U . Show that the sets {RU } and
{LU } are bases of uniformities on G, and prove that these uniformities
are different, but define the same topology.