cluster point: meaning, definition, pronunciation and examples
C2 / ProfessionalAcademic / Technical
Quick answer
What does “cluster point” mean?
A point in mathematics, specifically in topology or analysis, where every neighborhood contains infinitely many points of a given set.
Audio
Pronunciation
Definition
Meaning and Definition
A point in mathematics, specifically in topology or analysis, where every neighborhood contains infinitely many points of a given set.
In a broader, non-technical sense, a central point around which other elements, events, or ideas are densely grouped or concentrated.
Dialectal Variation
British vs American Usage
Differences
No significant difference in meaning or usage. The term is purely technical and used identically in both mathematical communities.
Connotations
Purely denotative in technical contexts; no connotative differences.
Frequency
Extremely low frequency in general language. Exclusively found in advanced mathematics textbooks and papers.
Grammar
How to Use “cluster point” in a Sentence
[Set] has a cluster point at [Number/Point].[Point] is a cluster point of the set [Set].Vocabulary
Collocations
Examples
Examples of “cluster point” in a Sentence
verb
British English
- [Not applicable as a verb]
American English
- [Not applicable as a verb]
adverb
British English
- [Not applicable as an adverb]
American English
- [Not applicable as an adverb]
adjective
British English
- [Not standard as an adjective. Use 'clustered'.]
American English
- [Not standard as an adjective. Use 'clustered'.]
Usage
Meaning in Context
Business
Not used.
Academic
Core term in real analysis and topology. Used to define compactness, closed sets, and continuity.
Everyday
Virtually never used. A metaphorical use might be: 'The city centre was a cluster point for protests.' This is highly non-standard.
Technical
Primary context. A cluster point of a sequence or set is a point such that every open set containing it contains infinitely many points from the sequence/set.
Vocabulary
Synonyms of “cluster point”
Strong
Weak
Vocabulary
Antonyms of “cluster point”
Watch out
Common Mistakes When Using “cluster point”
- Using 'cluster point' in everyday language.
- Confusing it with 'centroid' or 'centre of mass'.
- Incorrectly stating that a cluster point must belong to the set itself (it need not).
FAQ
Frequently Asked Questions
In most modern mathematical contexts (e.g., real analysis, general topology), the terms 'cluster point', 'accumulation point', and 'limit point' are used synonymously. Always check the specific definition in your textbook.
No. A finite set cannot have a cluster point because any point will have a neighbourhood small enough to exclude all other points of the finite set, contradicting the 'infinitely many points' requirement.
You analyse the set's adherence or closure. Often, you look for points where the set is 'dense' or where sequences within the set converge. For example, the cluster points of the set {1/n : n ∈ ℕ} is {0}.
Extremely rarely. Any non-mathematical use is a deliberate metaphor borrowed from the technical term and would be considered very specialised or literary.
A point in mathematics, specifically in topology or analysis, where every neighborhood contains infinitely many points of a given set.
Cluster point is usually academic / technical in register.
Cluster point: in British English it is pronounced /ˈklʌstə pɔɪnt/, and in American English it is pronounced /ˈklʌstɚ pɔɪnt/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “[None specific to this term]”
Learning
Memory Aids
Mnemonic
Imagine stars **clustering** densely around a specific **point** in the night sky. In maths, other points of the set swarm infinitely close to that central cluster point.
Conceptual Metaphor
A magnet attracting iron filings (the points of the set).
Practice
Quiz
Which of the following is a TRUE statement about a cluster point of a set S?