Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Assuming dependent choice, a nonempty topological space is uniformizable if and only if it is completely regular
Statement
Assuming dependent choice, a nonempty topological space is uniformizable if and only if it is completely regular.
Facts & Assumptions
Given: A nonempty topological space and dependent choice.
Under dependent choice, uniformizable spaces are completely regular (Assuming dependent choice, every uniformizable space is completely regular).
A completely regular topology is induced by its gauge of continuous pseudometrics (The topology of a nonempty completely regular space is induced by the gauge of its continuous -valued pseudometrics).
Uniformizable means induced by some uniformity (Uniformizable and separated-uniformizable topological spaces).
Proof
The forward implication is [L1].
The gauge supplied by [L2] is a uniformity inducing the given topology, so the reverse implication is [L2] and [L3].
The two implications prove the equivalence under dependent choice.
Depends on
- Assuming dependent choice, every uniformizable space is completely regular
- The topology of a nonempty completely regular space is induced by the gauge of its continuous $[0,1]$-valued pseudometrics
- Uniformizable and separated-uniformizable topological spaces
- The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
Used by
Dependency tree · two levels
22 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)
- M. Kunzinger, General Topology (standard reference, not scraped)
- M. Megrelishvili, Lecture Notes in Topological Groups (standard reference, not scraped)