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On a nonempty set, entourages and uniform covers give equivalent definitions of a uniform space in ZF, and under dependent choice they are also equivalent to gauges of pseudometrics
Statement
In ZF, on a nonempty set, the entourage and uniform-cover formulations determine each other. Assuming dependent choice, they are also equivalent to the formulation by gauges of pseudometrics.
Facts & Assumptions
Given: A uniform structure on a nonempty set in any one of the named formulations.
Entourage uniformities and uniform-cover structures determine each other in ZF (On a nonempty set, entourage uniformities and uniform-cover structures determine one another).
Under dependent choice every entourage uniformity is generated by a gauge of uniformly continuous pseudometrics (Assuming dependent choice, every entourage uniformity is generated by a gauge of uniformly continuous pseudometrics).
A gauge itself generates an entourage uniformity (A gauge of pseudometrics and, on a nonempty set, the uniformity it generates).
Proof
The equivalence between entourages and covers is exactly [L1] and uses no choice principle.
Assuming dependent choice, [L2] sends an entourage uniformity to a gauge and [L3] sends every gauge back to an entourage uniformity.
Thus the first two formulations are equivalent in ZF and all three are equivalent under the displayed assumption.
Depends on
- On a nonempty set, entourage uniformities and uniform-cover structures determine one another
- Assuming dependent choice, every entourage uniformity is generated by a gauge of uniformly continuous pseudometrics
- A gauge of pseudometrics and, on a nonempty set, the uniformity it generates
- The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 53 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- J. Wodzicki, Uniform Structure (standard reference, not scraped)
- Encyclopedia of Mathematics, Uniform space (standard reference, not scraped)
- M. Kunzinger, General Topology (standard reference, not scraped)