isolated set
LowTechnical/Academic
Definition
Meaning
A set of points in mathematics that are separated from all other points in the space, meaning each point in the set has a neighbourhood containing no other points of the set.
In broader contexts, it can refer to any collection of items, data points, or entities that are distinct and separated from a larger group or system, often implying a lack of connection or interaction.
Linguistics
Semantic Notes
Primarily a term from mathematical analysis and topology. The concept hinges on the idea of separation and discreteness. In non-technical use, it is often metaphorical, borrowing the mathematical sense of being separate and distinct.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. The term is technical and used identically in both varieties.
Connotations
Neutral, technical connotation in both varieties.
Frequency
Equally low frequency in both academic and technical registers.
Vocabulary
Collocations
Grammar
Valency Patterns
The [noun phrase] constitutes an isolated set.An isolated set of [plural noun] was identified.We prove that [set] is an isolated set.Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “[No common idioms for this technical term]”
Usage
Context Usage
Business
Rare. Might be used metaphorically in data analysis, e.g., 'The outliers formed an isolated set in the sales data.'
Academic
Primary context. Used in mathematics, computer science, and engineering papers dealing with topology, analysis, or graph theory.
Everyday
Extremely rare. Would likely be paraphrased, e.g., 'a separate group' or 'a bunch of things on their own.'
Technical
Common in technical fields like pure mathematics, theoretical computer science, and systems engineering where formal set theory is applied.
Examples
By Part of Speech
verb
British English
- The theorem allows us to isolate a set of critical vertices.
- We must first isolate the set of unstable parameters.
American English
- The algorithm isolates a set of potential matches.
- Researchers isolated the set of genes responsible for the trait.
adverb
British English
- [Not standardly used as an adverb for this noun phrase]
American English
- [Not standardly used as an adverb for this noun phrase]
adjective
British English
- The isolated set property is crucial for the proof.
- We examined the behaviour of the isolated set solution.
American English
- An isolated set condition must be satisfied.
- The model produced an isolated set result.
Examples
By CEFR Level
- [Too technical for A2 level]
- In the diagram, the three dots form an isolated set.
- The data showed a small, isolated set of errors.
- The proof relies on showing that the critical points constitute an isolated set.
- An isolated set of observations challenged the initial hypothesis.
- In topology, a finite set within a metric space is always an isolated set.
- The study focused on the dynamics near an isolated set of equilibrium points.
Learning
Memory Aids
Mnemonic
Think of islands (isolated) on a map. Each island is a point, and the collection of all islands is your 'isolated set'—each is separate from the others by water (the neighbourhood with no other points).
Conceptual Metaphor
ISLANDS IN AN OCEAN (points are islands, the space around them is the ocean ensuring separation).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid translating 'set' as 'набор' in this context; the correct mathematical term is 'множество'.
- Do not confuse with 'изолированный' meaning 'insulated' (as in wires); here it strictly means 'separated/standalone'.
Common Mistakes
- Using 'isolated set' to mean just 'a set that is alone' without the precise mathematical condition of each point having a neighbourhood excluding other set points.
- Confusing it with a 'closed set' or a 'bounded set', which are different properties.
Practice
Quiz
What is the defining property of an isolated set in a topological space?
FAQ
Frequently Asked Questions
No, an isolated set can be infinite. For example, the set of integers is an isolated set within the space of real numbers with the standard topology, as each integer has a neighbourhood (like an interval around it) containing no other integers.
Yes, but it is rare and typically a metaphorical extension of the mathematical concept. It might appear in fields like data science ('an isolated set of outliers') or sociology ('an isolated set of communities'), but the precise mathematical definition is usually relaxed.
An 'isolated point' is a single point that is separate from the rest of the set it belongs to. An 'isolated set' is a collection where *every* point in the collection is an isolated point with respect to that same collection.
Yes, vacuously. The condition for an isolated set is that for every point in the set, there exists a neighbourhood containing no other points of the set. Since there are no points in the empty set, the condition holds true for all (zero) of them.