Uniform space in the entourage formulation

Depends on

• Filter on a set
• The diagonal $\Delta_X \subseteq X \times X$, the diagonal map $\delta_X$, and the pairing $\langle f, g \rangle$ of two maps

Used by

• A gauge of pseudometrics and, on a nonempty set, the uniformity it generates Definition
• A uniformity with a countable entourage base Definition
• Cauchy filter in a uniform space Definition
• Separated uniformity: the intersection of all entourages is the diagonal Definition
• The left and right uniformities of a topological group Definition
• The pointwise and uniform-convergence uniformities on a function set Y^X Definition
• The upper and Roelcke uniformities generated from the left and right uniformities of a topological group Definition
• Totally bounded uniform space Definition
• Uniformly continuous map between uniform spaces Definition
• A metric on a nonempty set generates an entourage uniformity whose induced topology and uniformly continuous maps are the usual metric notions, and this uniformity is separated Lemma
• A normal sequence of entourages yields a uniformly continuous pseudometric with controlled dyadic balls Lemma
• Assuming dependent choice, every entourage admits a normal symmetric sequence subordinate to it Lemma
• Every convergent filter on a uniform space is Cauchy Lemma
• Every uniformity has a base of symmetric entourages Lemma
• Every uniformizable space is regular Lemma
• On a nonempty set, entourage uniformities and uniform-cover structures determine one another Lemma
• The standard entourages on minimal Cauchy filters form a separated uniformity Lemma
• Total boundedness passes to a uniform space with a dense uniformly continuous image Lemma
• A nonempty compact Hausdorff space carries exactly one compatible uniformity Theorem
• The sets containing an entourage ball about each of their points form a topology Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. Wodzicki, Uniform Structure (standard reference, not scraped)
• Encyclopedia of Mathematics, Uniform space (standard reference, not scraped)