ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedaudited 2026-09-22

How 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.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Development stars form a countable local base

Example

Let $(X, d)$ be a metric space (Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric) with its metric topology (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement). For $n \in N$ let $G_{n} : = {B (x, 2^{- n}) : x \in X},$ the cover by open balls of radius $2^{- n}$ (Open ball, closed ball and sphere in a metric space). The example computes the stars $St (x, G_{n})$ and verifies that $(G_{n})$ is a development, so that every metric space is a Moore space in the sense of Moore spaces and developments.

Facts & Assumptions

Given: A metric space $(X, d)$ , its balls $B (x, r)$ , and the covers $G_{n} = {B (x, 2^{- n}) : x \in X}$ .

[F1]

Star: $St (x, G_{n}) = ⋃ {B (z, 2^{- n}) : x \in B (z, 2^{- n})}$ , and each $G_{n}$ is an open cover because $x \in B (x, 2^{- n})$ and every metric ball is open (Refinements, locally finite families, point-finite families, and star refinements, Open ball, closed ball and sphere in a metric space, Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed, claim 1).

[L1]

Metric axioms: $d (x, y) = 0$ if and only if $x = y$ , symmetry, and the triangle inequality $d (x, y) \leq d (x, z) + d (z, y)$ (Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric).

[L2]

A set is open in the metric topology exactly when each of its points has a ball inside it, and balls are open (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed, claim 1).

[L3]

A space carrying its metric topology is metrizable, and every metrizable space is $T_{3}$ , hence regular $T_{1}$ (Metrizable space: a topological space whose topology is induced by some metric; metrizability is topological, the metric is not, In a metric space every closed set is a zero set and a $G_{δ}$ , and the distance function separates a point from a closed set, so every metrizable space is Tychonoff and perfectly normal, claim 4).

[L4]

For every $ε > 0$ some integer $m \geq 1$ satisfies $1 / m < ε$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

[L5]

A development is a sequence of open covers whose stars refine every open neighbourhood at each point; a Moore space is regular $T_{1}$ and developable. These stars give a countable local base, also for neighbourhoods that are not open (Moore spaces and developments, Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open).

Verification

technique · direct

1.1

For every $x$ and $n$ one has $B (x, 2^{- n}) \subseteq St (x, G_{n}) \subseteq B (x, 2^{1 - n})$ : the first inclusion holds because $x \in B (x, 2^{- n})$ ; for the second, if $y \in B (z, 2^{- n})$ with $x \in B (z, 2^{- n})$ then $d (x, y) \leq d (x, z) + d (z, y) < 2^{- n} + 2^{- n} = 2^{1 - n}$ by [L1].

givenF1L1

2.1

Hence $(G_{n})$ is a development. Let $U$ be open and $x \in U$ (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open). By [L2] there is $ε > 0$ with $B (x, ε) \subseteq U$ ; take $m \geq 1$ with $1 / m < ε$ by [L4] and set $n = m + 1$ . Induction gives $2^{m} \geq m$ , so $2^{1 - n} = 2^{- m} \leq 1 / m < ε$ ; then step 1.1 gives $St (x, G_{n}) \subseteq B (x, 2^{1 - n}) \subseteq B (x, ε) \subseteq U$ .

step 1.1L2L4L5

3.1

Consequently every metric space is developable, and [L3] makes it regular $T_{1}$ , so it is a Moore space; the star family ${St (x, G_{n}) : n \in N}$ is the countable local base at $x$ supplied by step 2.1: each star is open as a union of open balls and contains $x$ , and every neighbourhood contains an open neighbourhood to which step 2.1 applies. If $X$ is empty, the covers are empty and the assertions about points are vacuous.

step 1.1step 2.1L2L3L5∎

Remarks

The star bound doubles the radius. The lower bound shows the star is not smaller than the ball of radius $2^{- n}$ , and the upper bound shows it is contained in the ball of radius $2^{1 - n}$ ; that two-to-one gap is exactly what makes the development property hold with the factor $2$ .
The same computation works with any null sequence of radii, the powers $2^{- n}$ being chosen only for definiteness.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

56 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

Dennis K. Burke, The Normal Moore Space Problem (standard reference, not scraped)