accumulation point
C2 (Specialist Technical)Technical/Formal
Definition
Meaning
A point in a topological space where every neighborhood contains infinitely many points from a given set.
In mathematics, a point that can be approximated arbitrarily closely by other points from a given set (excluding the point itself). In general contexts, a focal point where things gather or converge over time.
Linguistics
Semantic Notes
Primarily a term in real analysis, topology, and complex analysis. It implies a limit-like property but distinct from a limit point in some topological contexts. The set must have infinitely many points near the accumulation point.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. The term is standardized in international mathematics.
Connotations
Purely technical, carries no regional connotative differences.
Frequency
Equally rare and specialized in both dialects, confined to university-level mathematics.
Vocabulary
Collocations
Grammar
Valency Patterns
accumulation point of [SET][SET] has an accumulation point at [NUMBER/POINT][SET]'s accumulation points are...Let x be an accumulation point of S.Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Usage
Context Usage
Business
Not used. Potential metaphorical use: 'The city centre is an accumulation point for talent and investment.'
Academic
Core term in real analysis, topology, and complex analysis courses.
Everyday
Virtually never used.
Technical
Precise definition crucial for proofs regarding closed sets, compactness, and continuity.
Examples
By Part of Speech
noun
British English
- The proof hinges on showing that every infinite bounded set in R has at least one accumulation point.
- Is zero an accumulation point for the set {1/n : n ∈ N}?
American English
- The concept of an accumulation point is fundamental to understanding the Bolzano-Weierstrass theorem.
- We need to classify all accumulation points of the given metric space.
Examples
By CEFR Level
- In mathematics, an accumulation point is where many points of a set gather closely.
- A set is closed if and only if it contains all of its accumulation points, a key theorem in topology.
- The sequence oscillates, but its range has only one accumulation point at the origin.
Learning
Memory Aids
Mnemonic
Think of a magnet (the point) attracting iron filings (other points) to gather infinitely close around it.
Conceptual Metaphor
A SOCIAL HUB: A place where members of a group (points of the set) constantly congregate in its immediate vicinity.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Do not confuse with 'точка сгущения' (condensation point), which may have a stricter definition.
- In some older Russian texts, 'предельная точка' (limit point) is used synonymously.
Common Mistakes
- Confusing an accumulation point with a limit of a sequence (a sequence has a limit, a set has accumulation points).
- Forgetting that the point itself does not need to be in the set.
- Assuming finite sets can have accumulation points (they cannot).
Practice
Quiz
Which of the following sets has 0 as an accumulation point?
FAQ
Frequently Asked Questions
In a T1 space (like metric spaces), yes, the terms are often used synonymously. In more general topology, definitions can vary slightly.
No, by definition. An isolated point has a neighbourhood containing no other points from the set, contradicting the requirement for an accumulation point.
An accumulation point requires other points from the set to be arbitrarily close. A boundary point only requires points from the set AND its complement to be close. All accumulation points are boundary points if not interior points.
It's essential for defining closed sets, compactness, continuity (via limits), and theorems like Bolzano-Weierstrass, which is foundational for calculus and analysis.