LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passverified 2026-08-09 (gpt-5.6-terra-codex-subscription)

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The topology of a nonempty completely regular space is induced by the gauge of its continuous $[0,1]$ -valued pseudometrics

Statement

The topology of a nonempty completely regular space is induced by the gauge of pseudometrics $p_f(x,y)=|f(x)-f(y)|$ , where $f:X\to[0,1]$ ranges over continuous maps.

Facts & Assumptions

Given: A nonempty completely regular space $X$ .

[L1]

Complete regularity separates a point from a closed set by a continuous $[0,1]$ -valued function (Completely regular spaces and Tychonoff ( $T_{3\frac{1}{2}}$ ) spaces, Intervals of $\mathbb{R}$ : the nine order-convex forms, nondegeneracy, and length).

[L2]

Such functions are continuous in the neighbourhood sense (Continuity of a map of topological spaces at a point and globally).

[L3]

A gauge generates a uniformity from finite simultaneous pseudometric balls (A gauge of pseudometrics and, on a nonempty set, the uniformity it generates).

[L4]

Absolute value is nonnegative, vanishes only at zero and is even (Basic properties of the absolute value), and it satisfies $|u+v|\le |u|+|v|$ (The triangle inequality).

Proof

technique · constructive

1.1

For each continuous $f:X\to[0,1]$ , direct substitution in [L4] shows that $p_f(x,y)=|f(x)-f(y)|$ is nonnegative, symmetric, zero on the diagonal and satisfies the triangle inequality, so it is a pseudometric; its balls about $x$ are original-open by [L2].

L2L4construct

1.2

Conversely, if $x\in U$ is original-open, apply [L1] to the closed set $X\setminus U$ to obtain $f$ with $f(x)=1$ and $f[X\setminus U]=\{0\}$ ; then the $p_f$ -ball of radius $1/2$ about $x$ lies in $U$ .

L1L3choose

2.1

Hence every gauge-open set is original-open.

step 1.1L3

3.1

Thus original-open and gauge-open sets contain one another, so the two topologies agree.

step 2.1step 1.2discharge-construct∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 73 results over 15 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)
M. Kunzinger, General Topology (standard reference, not scraped)
M. Megrelishvili, Lecture Notes in Topological Groups (standard reference, not scraped)

The topology of a nonempty completely regular space is induced by the gauge of its continuous [0,1][0,1][0,1]-valued pseudometrics