Notes about uniform spaces from Preuss, . . ..

Definition. Uniformity ω is a filter on X × X such that: U1) W ∈ ω ⇒ △ = {(x, x) : x ∈ X} ⊂ W U2) W ∈ ω ⇒ W^{−1} ∈ ω U3) W ∈ ω ⇒ (∃W^{∗} ∈ ω)(W^{∗})^{2} ⊂ W (X, ω) - uniform space Elements of ω - entourages

For each V ∈ ω A ⊂ X is called V -small if A × A ⊂ V.

If (X, d) is a metric (pseudometric) space and V_{ε} = {(x, y) : d(x, y) < ε} for each ε > 0, then B = {V_{ε}; ε > 0} is a base for uniformity ω_{d} on X.

Proclaim. Let (X, ω) be a uniform space. For each x ∈ X and each V ∈ ω let V (x) = {y : (x, y) ∈ V }. Then X_{ω} = {O ⊂ X : there is some V ∈ ω with V (x) ⊂ O} is a topology on X.

Definition. Let (X, ω), (X′, ω′) be a uniform spaces. A map f: X → X′ is called uniformly continuous provided that one ofthe following two equivalent conditions is satisfied:

(i) For each W′ ∈ ω′ there is some W ∈ ω such that (x, y) ∈ W ⇒ (f(x), f(y) ∈ W′.

(ii) (f × f)^{−1}[W′] ∈ ω for each W′ ∈ ω′, where f × f : (x, y) ↦ (f(x), f(y)).

initial uniformity, product space, supremum of family of uniformities on X

Theorem. For each uniformity ω on a set X there is a family (d_{V})_{v∈ω} if pseudometrics on X such that for the corresponding uniformities D_{v} the following is true: ω = sup{D_{v} : V ∈ ω}.

Theorem. A topological space (X, X) is uniformizable (i.e. there is a uniformity ω on X such that X_{ω} = X) if and only if it is completely regular.

Definition. A uniform space (X, ω) is called separated iff ⋃_{W ∈ω} W = △.

A uniform space (X, ω) is called pseudometrizable (metrizable) provided that there is a pseudometric (metric) d on X such that ω_{d} = ω.

Theorem. An uniform space is pseudometrizable if and only if it has countable base.

An uniform space is metrizable if and only if it is separable and has countable base.

Definition. A filter F on a uniform space (X, ω) is called a Cauchy filter provided that for each W ∈ ω there is some F ∈ F such that F × F ⊂ W.

A uniform space (X, ω) is called complete provided that each Cauchy filter on (X, ω) is convergent (in (X, X_{ω})).

Theorem. Let (X, ω) be a separated uniform space. Then there is a dense embedding into a complete separated uniform space. Moreover, it has properties of reflection and is determined uniquely. It is called complete hull.

Definition. A uniform space (X, ω) is called totally bounded provided that one of the following equivalent conditions is satisfied:

(i) For each V ∈ ω there is a finite cover of X by V -small sets. (ii) For each W ∈ ω there is a finite subset E of X such that W [E] = X,

where W [E] = ⋃_{x∈E} W (x).

(iii) Each ultrafilter on X is Cauchy filter. Theorem. A uniform space (X, ω) is compact (i.e. (X, X_{ω}) is compact) if and only if it is complete and totally bounded.