M6140 Topology Exercises - 9th Week (2020)
1 Uniform Spaces
Exercise 1. Suppose that U, V are uniformities on X and U1, V1 are their bases, respectively. Prove
that U ⊆ V if and only if for each U ∈ U1 there exists V ∈ V1 such that V ⊆ U.
Exercise 2. Show that a non-empty set V ⊆ X × X is a basis of some uniformity if and only if
(i) ∆X ⊆ U for each U ∈ V,
(ii) for each U, V ∈ V there exists W ∈ V such that W ⊆ U ∩ V ,
(iii) for each U ∈ V there exists V ∈ V such that V −1 ⊆ U, and
(iv) for each U ∈ V there exists V ∈ V such that V ◦ V ⊆ U.
Exercise 3. Let (X, U), (Y, V) be uniform spaces with bases U1, V1, respectively. Prove that a
mapping f : X → Y is uniformly continuous if and only if for each V ∈ V1 there exists U ∈ U1 such
that U ⊆ (f × f)−1(V ).
Definition 1. A non-empty set V ⊆ X × X is called a subbasis of some uniformity if the set of its
finite intersections is a basis of the uniformity.
Exercise 4. Show that a non-empty set S ⊆ X × X is a subbasis of some uniformity if
(i) ∆X ⊆ U for each U ∈ S,
(ii) for each U ∈ S there exists V ∈ S such that V −1 ⊆ U, and
(iii) for each U ∈ S there exists V ∈ S such that V ◦ V ⊆ U.
Exercise 5. Prove that a mapping between metric spaces is uniformly continuous in the sense of
metric spaces if and only if it is uniformly continuous in the sense of uniform spaces.
Exercise 6. Let G be a topological group. Show that
{(x, y) ∈ G × G | x · y−1
∈ V } | V is a neighborhood of 1.
is a basis of some uniformity on G.
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