Locally Discrete Collection¶
A family of subsets of a topological space for which every point has an open neighborhood meeting at most one family member.
Core Idea¶
A locally discrete collection is a family of subsets of a topological space for which every point has an open neighborhood meeting at most one family member. It is a condition on the chosen family, not on whether all points of the space are isolated. Bing's 1951 discrete collection definition uses an equivalent closure condition; the neighborhood wording here is a mathematical derivation, not his verbatim sentence.[^ref-236cd1cf6716]
Scope of Application¶
Bing constructs discrete open families one layer at a time in the necessity proof of his metrization theorem for metric spaces. His Example D supplies a contrasting discrete open refinement in a regular separable lower-limit-line space that he shows is nonmetrizable. The same local family test applies in both, while metrizability depends on further conditions beyond the existence of one such family.[^ref-236cd1cf6716]
Clarity¶
For every point, including one outside the union of the members, find an open neighborhood meeting no more than one distinct member. Pairwise disjointness alone cannot pass this test: disjoint intervals may accumulate at a limit point where every neighborhood meets many of them. Local finiteness allows several members nearby; local discreteness permits at most one. A family may pass without covering the space or being a base.[^ref-236cd1cf6716]
Manages Complexity¶
The local test controls a possibly large family by choosing a witness neighborhood at each point. Bing's equivalent closure statement requires pairwise disjoint member closures and the union of closures of every subcollection to be closed. A point in two closures would force every neighborhood to meet two members; the closed-union condition prevents distant accumulation outside the chosen family. The equivalence is derived from his stated definition rather than silently attributed as his exact wording.[^ref-236cd1cf6716]
Abstract Reasoning¶
Fix an arbitrary point and count how many family members any suitable open neighborhood meets. Zero or one passes there; if every neighborhood meets two, the family fails. Apply this pointwise to the whole space. When a proof uses a countable sequence of discrete families, test each layer separately and then check the additional coverage or local-basis property. Their union need not itself be one discrete family.[^ref-236cd1cf6716]
Knowledge Transfer¶
The definition transfers from metric to nonmetric topology because it requires open neighborhoods, not distances. Bing's two constructions witness that breadth. The live Topological Space entry is a strict ambient prerequisite: remove the topology and the neighborhood condition is undefined, while a space can exist without a designated family. The typed edge is composition/presupposes, not a claim that the family is a kind of space. The broad Discreteness Prime is a thematic neighbor, not an established strict direct parent.[^ref-236cd1cf6716]
Example¶
In Bing's metric-space construction, choose one fixed discrete open family from his scale-indexed sequence. Mapped back: the metric space supplies the ambient topology, the chosen layer supplies members, and its discreteness supplies an open neighborhood meeting at most one member at each point. The countable sequence and basis property used for metrizability are separate from this one-family identity.[^ref-236cd1cf6716]
In Bing's Example D, an open cover of his nonmetrizable lower-limit line has an open refinement he identifies as discrete. Mapped back: that line supplies a different ambient topology, the chosen refinement supplies members, and its discrete status supplies the same pointwise witnesses. The example shows that even a discrete refinement does not by itself prove metrizability or a countable discrete base.[^ref-236cd1cf6716]
Relationships to Other Abstractions¶
Current abstraction Locally Discrete Collection Domain-specific
Parents (1) — more general patterns this builds on
-
Locally Discrete Collection presupposes Topological Space Domain-specific
A locally discrete collection requires an ambient topology to define each point's open neighborhood and test encounters with family members.
Hierarchy paths (5) — routes to 3 parentless roots
- Locally Discrete Collection → Topological Space → Closure
- Locally Discrete Collection → Topological Space → Set and Membership
- Locally Discrete Collection → Topological Space → Topology
- Locally Discrete Collection → Topological Space → Intersection → Set and Membership
- Locally Discrete Collection → Topological Space → Union → Set and Membership
Neighborhood in Abstraction Space¶
Locally Discrete Collection sits in a sparse region of the domain-specific corpus (76th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Point-Set Topology Foundations (17 abstractions)
Nearest neighbors
- Interior — 0.85
- Open Set — 0.84
- Locally Closed Subset — 0.83
- Subspace Topology — 0.83
- Collectionwise Normal Space — 0.83
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
Discrete topology isolates every point; a locally discrete family need not live in one. Pairwise disjointness ignores external accumulation. Local finiteness permits more than one member nearby. A sigma-discrete base is a countable union of discrete layers with a basis property, not one discrete family. Metrizability requires the full hypotheses of Bing's theorem, not merely one positive family example.[^ref-236cd1cf6716]
References¶
[^ref-236cd1cf6716]: R. H. Bing, “Metrization of Topological Spaces,” Canadian Journal of Mathematics 3 (1951), pp. 175–186, especially p. 176 (discrete collection definition), p. 179 (Theorem 3 necessity construction), and p. 181 (Example D); original journal PDF. https://www.cambridge.org/core/services/aop-cambridge-core/content/view/48C1A50A9E249D05BD7054529F93BAA1/S0008414X00030923a.pdf/metrization-of-topological-spaces.pdf