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

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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

1.1givenF1L1

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].

2.1step 1.1L2L4L5

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 $St (x, G_{n}) \subseteq B (x, 2^{1 - n}) \subseteq B (x, ε) \subseteq U$ .

3.1step 1.1step 2.1L2L3L5∎

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.

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)