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The topology of a nonempty completely regular space is induced by the gauge of its continuous -valued pseudometrics
Statement
The topology of a nonempty completely regular space is induced by the gauge of pseudometrics , where ranges over continuous maps.
Facts & Assumptions
Given: A nonempty completely regular space .
Complete regularity separates a point from a closed set by a continuous -valued function (Completely regular spaces and Tychonoff () spaces, Intervals of : the nine order-convex forms, nondegeneracy, and length).
Such functions are continuous in the neighbourhood sense (Continuity of a map of topological spaces at a point and globally).
A gauge generates a uniformity from finite simultaneous pseudometric balls (A gauge of pseudometrics and, on a nonempty set, the uniformity it generates).
Absolute value is nonnegative, vanishes only at zero and is even (Basic properties of the absolute value), and it satisfies (The triangle inequality).
Proof
For each continuous , direct substitution in [L4] shows that is nonnegative, symmetric, zero on the diagonal and satisfies the triangle inequality, so it is a pseudometric; its balls about are original-open by [L2].
Conversely, if is original-open, apply [L1] to the closed set to obtain with and ; then the -ball of radius about lies in .
Hence every gauge-open set is original-open.
Thus original-open and gauge-open sets contain one another, so the two topologies agree.
Depends on
- Completely regular spaces and Tychonoff ($T_{3\frac{1}{2}}$) spaces
- A gauge of pseudometrics and, on a nonempty set, the uniformity it generates
- Continuity of a map of topological spaces at a point and globally
- Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length
- Basic properties of the absolute value
- The triangle inequality
Used by
- Assuming dependent choice, a nonempty topological space is separated-uniformizable if and only if it is Tychonoff Corollary
- Assuming dependent choice, a nonempty topological space is uniformizable if and only if it is completely regular Theorem
- Under dependent choice and the ultrafilter lemma, uniformly continuous maps to compact Hausdorff spaces extend uniquely over the Samuel compactification Theorem
Dependency tree · two levels
28 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)