Show that the families defined in Examples 5.3.1, 5.3.2, 5.3.3, and 5.3.4 are in fact uniformities. Example 5.3.1. Given a prime number p, the p-adic uniformity on Z is generated by the entourages of the form:Dn = {(x, y) ∈ Z × Z : x ≡ y mod pn}, n ∈ N \ {0}. Example 5.3.2. The additive uniformity on a topological vector space E, has a basis formed by entourages of the form: {(x, y) ∈ E × E : x − y ∈ V }, where V is a neighborhood of the zero vector of E. Example 5.3.3. The left uniformity UL(G) on a topological group G, has a basis formed by entourages of the form: {(x, y) ∈ G × G : x−1y ∈ V },where V is a neighborhood of the identity element of G. Analogously, the right uniformity UR(G) is generated by the entourages of the form: {(x,y)∈G×G:xy−1 ∈V}, where V is a neighborhood of the identity element of G. Obviously, if G is an Abelian group, UL(G) and UR(G) coincide. Example 5.3.4. The (pseudo)metric uniformity on a (pseudo)metric space (X, d) is generated by the entourages Vεd := {(x, y) ∈ X × X : d(x, y) < ε}, ε > 0.
Show that the families defined in Examples 5.3.1,
5.3.2, 5.3.3, and 5.3.4 are in fact uniformities.
Example 5.3.1. Given a prime number p, the p-adic uniformity on Z
is generated by the entourages of the form:
Dn = {(x, y) ∈ Z × Z : x ≡ y mod pn}, n ∈ N \ {0}.
Example 5.3.2. The additive uniformity on a topological
{(x, y) ∈ E × E : x − y ∈ V }, where V is a neighborhood of the zero vector of E.
Example 5.3.3. The left uniformity UL(G) on a topological group G, has a basis formed by entourages of the form:
{(x, y) ∈ G × G : x−1y ∈ V },
where V is a neighborhood of the identity element of G. Analogously,
the right uniformity UR(G) is generated by the entourages of the form: {(x,y)∈G×G:xy−1 ∈V},
where V is a neighborhood of the identity element of G. Obviously, if G is an Abelian group, UL(G) and UR(G) coincide.
Example 5.3.4. The (pseudo)metric uniformity on a (pseudo)metric space (X, d) is generated by the entourages
Vεd := {(x, y) ∈ X × X : d(x, y) < ε}, ε > 0.
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