strong accumulation point: meaning, definition, pronunciation and examples
Very Low (Specialist Technical Term)Formal, Academic, Technical (Mathematics)
Quick answer
What does “strong accumulation point” mean?
A point in a topological space such that every neighborhood of that point contains uncountably many points of the set.
Audio
Pronunciation
Definition
Meaning and Definition
A point in a topological space such that every neighborhood of that point contains uncountably many points of the set.
In mathematical analysis, a point where not only does the set accumulate (adherent point), but it does so with the cardinality of the continuum (uncountably many points) in every neighborhood.
Dialectal Variation
British vs American Usage
Differences
No lexical differences. Concept and terminology identical in UK/US mathematical communities.
Connotations
Pure technical precision; no cultural connotations.
Frequency
Extremely rare outside advanced mathematics textbooks and research papers.
Grammar
How to Use “strong accumulation point” in a Sentence
[Set S] has [a strong accumulation point] at [point x].[Point x] is [a strong accumulation point] of [set S].Vocabulary
Collocations
Examples
Examples of “strong accumulation point” in a Sentence
adjective
British English
- The set's strong accumulation point behaviour was critical to the proof.
American English
- The strong accumulation point property failed to hold in the general case.
Usage
Meaning in Context
Business
Never used.
Academic
Exclusively used in advanced mathematics, specifically real analysis, topology, and descriptive set theory.
Everyday
Never used.
Technical
The primary and only context. Used with precise definitions in proofs and theorems.
Vocabulary
Synonyms of “strong accumulation point”
Neutral
Weak
Vocabulary
Antonyms of “strong accumulation point”
Watch out
Common Mistakes When Using “strong accumulation point”
- Confusing it with a limit point/accumulation point (which only requires one point, not uncountably many).
- Using it in non-mathematical contexts.
- Thinking 'strong' implies metric strength or force.
FAQ
Frequently Asked Questions
Yes. By definition, if every neighbourhood has uncountably many points, it certainly has at least one (other) point, so it is an accumulation point.
An accumulation point requires every neighbourhood to contain at least one other point of the set. A strong accumulation point requires every neighbourhood to contain uncountably many points of the set.
Primarily in real analysis, point-set topology, and descriptive set theory when discussing the cardinality of limit points.
No. If the set is countable, it cannot have an uncountable subset. Therefore, no neighbourhood can contain uncountably many points from that set, precluding a strong accumulation point.
A point in a topological space such that every neighborhood of that point contains uncountably many points of the set.
Strong accumulation point is usually formal, academic, technical (mathematics) in register.
Strong accumulation point: in British English it is pronounced /ˌstrɒŋ əˌkjuːmjəˈleɪʃən ˌpɔɪnt/, and in American English it is pronounced /ˌstrɔŋ əˌkjumjəˈleɪʃən ˌpɔɪnt/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
STRONG = So That Really Oodles 'N' Gobs (of points cluster there). Not just a few, but uncountably many.
Conceptual Metaphor
A magnet so powerful it attracts not just iron filings (points), but an inseparable, continuous blob of them from every direction.
Practice
Quiz
Which statement best defines a strong accumulation point?