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A Moore space looks metric one scale at a time. It is regular and has a development: a sequence of open covers whose stars around each point eventually fit inside every neighbourhood. That local approximation still needs enough global separation. Bing’s theorem supplies the missing bridge: a Moore space is metrizable exactly when it is collectionwise normal, so every discrete family of closed sets can be expanded into separated open neighbourhoods.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. Define a development through stars
For an open cover G and point x, the star St(x,G) is the union of all members of G containing x. A sequence G1,G2,… is a development when these stars form a neighbourhood base at each point.
Equivalently, for every open U containing x, some n satisfies St(x,Gn)⊂U. The index n acts like a scale, although no numerical distance has yet been supplied.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. See why a Moore space is more than first countable
A Moore space is a regular developable space, with the usual T1 convention stated where necessary. A development coordinates local bases across the whole space through common covers.
First countability gives each point its own countable local base. It does not require those bases to arise from one shared sequence of open covers.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Work the real-line development
On the real line, let Gn be the cover by all open intervals of radius 1/n centred at real numbers. Every interval in Gn that contains x lies within distance 2/n of x, so St(x,Gn)⊂(x−2/n,x+2/n).
| Scale | Cover elements | Star around x | Use |
|---|---|---|---|
| n=1 | Radius-1 intervals | Inside (x−2,x+2) | Coarse neighbourhood |
| n=10 | Radius-0.1 intervals | Inside (x−0.2,x+0.2) | Finer neighbourhood |
| Large n | Radius-1/n intervals | Inside (x−2/n,x+2/n) | Eventually fits every open neighbourhood |
The calculation demonstrates development; the familiar Euclidean metric already supplies regularity and the final compatible distance.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Define collectionwise normality
A discrete family of closed sets is one for which every point has a neighbourhood meeting at most one member. A space is collectionwise normal when every such family can be assigned pairwise disjoint open expansions, one around each closed set.
| Property | What it separates | Requirement |
|---|---|---|
| Normal | Two disjoint closed sets | Disjoint open neighbourhoods |
| Collectionwise normal | An arbitrary discrete family of closed sets | Pairwise disjoint open expansions |
| Regular | A point and a disjoint closed set | Separated neighbourhood control |
For a two-member family, normality supplies the required separation. The distinction emerges for large discrete families.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. State Bing’s Moore metrisation theorem
Bing proved that a Moore space is metrizable if and only if it is collectionwise normal. The forward direction follows from standard metric-space separation; the reverse direction turns the development and simultaneous separation into a σ-discrete base.
Once that base is obtained, Bing’s general metrisation theorem converts regularity plus a σ-discrete base into a compatible metric.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Follow the proof architecture
Begin with one cover in the development. Refine it by closed families arranged in discrete layers. Collectionwise normality expands each discrete closed layer into a discrete open family that still refines the original cover.
Repeat across the countable development. The resulting countable union of discrete open families is a base because the stars eventually fit inside every prescribed neighbourhood.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Transfer the test to a merely normal space
Normality separates one pair of closed sets at a time. It does not in ZFC automatically separate every discrete closed family, so it cannot simply replace collectionwise normality in Bing’s theorem.
The changed-context check is historical as well as logical: the statement that every normal Moore space is metrizable is independent of the usual ZFC axioms. Do not present normality alone as an unconditional substitute.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the three metrisation routes
Use Bing metrisation and σ-discrete bases for the final base criterion. Use Nagata-Smirnov metrisation and locally finite bases for a parallel layered-base route.
The guide to Urysohn’s lemma and normal-space separation explains two-set separation, which is weaker than the family-wise condition used here.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Know the limits and common misconceptions
A development is not merely a nested list of covers; its stars must form local bases. A developable space need not be metrizable unless the required regularity and collectionwise separation are present.
Collectionwise Hausdorff and collectionwise normal are not interchangeable. The former separates discrete families of points; the latter handles discrete families of closed sets.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Use a proof and source checklist
State separation conventions, display the covers, calculate stars, verify the development property, define the discrete closed family, show the open expansions and explain how the resulting open layers form a base.
Bing’s original Metrization of Topological Spaces introduces collectionwise normality and proves the Moore-space criterion. The Texas State University archive Bing Metrization Theorem records the theorem’s statement and historical context.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Does every metric space have a development?
Yes. Covers by sufficiently small metric balls give shrinking stars after choosing a suitable scale sequence.
Is every normal Moore space metrizable?
That statement is independent of ZFC; Bing’s unconditional criterion uses collectionwise normality.
Does a development itself define a unique metric?
No. It supplies topological scale information, and multiple compatible metrics may result.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue with a changed-space check
Replace the radius-1/n covers of the real line by radius-1/√n covers. Prove that the stars still form a local base. Then explain why using radius one at every n fails.
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