locally finite set: meaning, definition, pronunciation and examples
C2Technical/Scientific/Academic
Quick answer
What does “locally finite set” mean?
In mathematics, a set with the property that every element intersects only finitely many other elements of the set.
Audio
Pronunciation
Definition
Meaning and Definition
In mathematics, a set with the property that every element intersects only finitely many other elements of the set.
A concept from set theory, topology, combinatorics, and group theory where a collection of sets, points, or elements has finite intersections within any neighbourhood of the space.
Dialectal Variation
British vs American Usage
Differences
No lexical differences. The term and its definition are identical in international mathematics literature. Spelling of "neighbourhood/neighborhood" may follow regional conventions in surrounding text.
Connotations
None; purely technical.
Frequency
Extremely low frequency in general language. Exclusively used in advanced mathematical discourse with identical frequency across regions.
Grammar
How to Use “locally finite set” in a Sentence
[The family/collection] is locally finite.There exists a locally finite [cover/refinement].A locally finite set of [points/intervals].Vocabulary
Collocations
Examples
Examples of “locally finite set” in a Sentence
adjective
British English
- The proof relies on constructing a locally finite open cover of the space.
American English
- A key property of the decomposition is that it is locally finite.
Usage
Meaning in Context
Business
Not used.
Academic
Central term in advanced mathematics papers, particularly in topology, set theory, and geometric group theory.
Everyday
Not used.
Technical
Precise definition used in proofs, lemma statements, and mathematical constructions.
Vocabulary
Synonyms of “locally finite set”
Strong
Neutral
Weak
Vocabulary
Antonyms of “locally finite set”
Watch out
Common Mistakes When Using “locally finite set”
- Confusing it with a 'finite set'. A locally finite set can be (and often is) infinite globally.
- Using it outside of a mathematical context.
- Incorrectly applying it to a set of numbers without a defined topological neighbourhood structure.
FAQ
Frequently Asked Questions
Yes. 'Locally finite' describes a local property. The entire set can be infinite, but each element or point only interacts with a finite number of others in its immediate neighbourhood.
'Finite' means the entire set has a limited number of elements. 'Locally finite' refers to an infinite set where, when you zoom in on any small area (neighbourhood), you only see a finite number of elements from the set relevant to that area.
Almost exclusively in advanced university-level mathematics textbooks and research papers, particularly in areas like topology, set theory, geometric group theory, and combinatorics.
It is most precisely a property of a *collection* of sets (a family) within a topological space. Informally, one might call the set of all points in such a collection a 'locally finite set', but the technical term applies to the family of subsets covering or partitioning the space.
In mathematics, a set with the property that every element intersects only finitely many other elements of the set.
Locally finite set is usually technical/scientific/academic in register.
Locally finite set: in British English it is pronounced /ˌləʊkəli ˈfaɪnaɪt ˈsɛt/, and in American English it is pronounced /ˌloʊkəli ˈfaɪnaɪt ˈsɛt/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of a city (the set) where every resident (a point) has only a finite number of neighbours within their immediate street (local neighbourhood).
Conceptual Metaphor
KNOWLEDGE AS SPACE. A 'locally finite set' metaphorically represents a complex structure where understanding is manageable in small, focused areas, even if the whole is vast.
Practice
Quiz
In which field is the term 'locally finite set' exclusively used?