isolated point
C2Technical/Academic
Definition
Meaning
A point in a topological space that has a neighbourhood containing no other points of the set.
A point that is separate, detached, or disconnected from others; something or someone that stands alone without connection to a larger group or system.
Linguistics
Semantic Notes
Primarily a mathematical term in topology and analysis; used metaphorically in other fields to describe something disconnected or exceptional.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning; identical usage in mathematical contexts.
Connotations
Slightly more likely to be used metaphorically in British academic writing.
Frequency
Equally rare in everyday speech; primarily academic/technical term in both varieties.
Vocabulary
Collocations
Grammar
Valency Patterns
[isolated point] of [set/space][set] contains [isolated point][point] is isolated in [space]Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “An isolated point in the data”
- “Like an isolated point in the system”
Usage
Context Usage
Business
Rare; might describe a single data point disconnected from trends.
Academic
Common in mathematics, physics, and engineering papers.
Everyday
Virtually never used in casual conversation.
Technical
Standard term in topology, complex analysis, and dynamical systems.
Examples
By Part of Speech
verb
British English
- The function isolates points where the derivative vanishes.
- We can isolate points of discontinuity.
American English
- The algorithm isolates points of high curvature.
- They isolated the critical points in the dataset.
adverb
British English
- The data points lie isolatedly in that region.
- Points occur isolatedly in the parameter space.
American English
- The measurements are distributed isolatedly.
- Events happened isolatedly in time.
adjective
British English
- The isolated point topology is quite specialised.
- We studied isolated point singularities.
American English
- The isolated point case requires separate treatment.
- This is an isolated point phenomenon.
Examples
By CEFR Level
- On the map, the village was just an isolated point in the mountains.
- The scientist marked an isolated point on the graph.
- In the dataset, that measurement appears as an isolated point with no correlation to others.
- The theorem proves that every finite set consists only of isolated points.
- The existence of an isolated point in the spectrum has significant implications for the operator's properties.
- If every point of a set is an isolated point, then the set must be discrete in the subspace topology.
Learning
Memory Aids
Mnemonic
Imagine a single dot floating alone in empty space—completely isolated from any other dots.
Conceptual Metaphor
DETACHMENT IS ISOLATION, CONNECTION IS CONTINUITY
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid translating as 'отдельная точка' when mathematical precision is needed—use 'изолированная точка'.
- Don't confuse with 'уединённая точка' which is less standard.
Common Mistakes
- Using 'isolated point' to mean 'remote location' in formal writing.
- Confusing with 'outlier' in statistics (related but distinct).
Practice
Quiz
In which field is 'isolated point' primarily a technical term?
FAQ
Frequently Asked Questions
Rarely; it's overwhelmingly a mathematical term, though occasionally used metaphorically in technical writing.
An outlier is a data point distant from others statistically; an isolated point is a topological concept about neighbourhoods.
Yes, such sets are called discrete sets in topology.
By exhibiting an open neighbourhood around it containing no other points from the set.