open cover
C2Technical (Mathematics); Formal/General
Definition
Meaning
A noun phrase describing a collection of open sets in a topological space whose union contains a given set.
In general use, it can refer to an arrangement or agreement (like insurance) that is broadly defined and without hidden restrictions.
Linguistics
Semantic Notes
As a single lexical unit, it is almost exclusively a term of art in mathematics (topology). In other contexts, it behaves as a free combination of adjective + noun ('an open cover').
Dialectal Variation
British vs American Usage
Differences
No significant difference in the technical mathematical term. In general language, 'cover' might be more common in UK English for insurance/financial contexts (e.g., 'insurance cover'), while US English might favor 'coverage'.
Connotations
Technical/mathematical connotation dominates. In general use, it suggests transparency and lack of restrictions.
Frequency
Extremely rare in everyday language; high frequency in advanced mathematical texts.
Vocabulary
Collocations
Grammar
Valency Patterns
[open cover] of [set/space][set] has an [open cover]Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “None specific to this term as a unit.”
Usage
Context Usage
Business
Rare. Could refer to transparent terms in an insurance policy: 'The policy provides open cover for all company vehicles.'
Academic
Primary usage: topology/analysis. 'The proof relies on finding a suitable open cover of the metric space.'
Everyday
Virtually never used as a fixed phrase. Interpreted literally: 'Please remove the open cover from the jar.'
Technical
Core technical term in mathematics.
Examples
By Part of Speech
adjective
British English
- The concept of an open cover is fundamental.
American English
- An open cover definition is introduced in Chapter 3.
Examples
By CEFR Level
- In mathematics, an open cover for a set is a family of open sets whose union contains it.
- The compactness of the interval can be demonstrated by showing that every open cover has a finite subcover.
- The marine insurance was written on an open cover, allowing for new shipments to be added automatically.
Learning
Memory Aids
Mnemonic
Imagine a blanket (cover) that is full of holes (open sets). This holey blanket is spread open to cover a complex shape (the set).
Conceptual Metaphor
PROTECTION/INCLUSION IS A PHYSICAL COVERING (technical); TRANSPARENCY IS BEING OPEN (general).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Do not translate as 'открытая обложка' (book cover). The mathematical term is 'открытое покрытие'. In general language, it's just adjective + noun.
Common Mistakes
- Using it as a verb phrase ('to open cover').
- Confusing it with the phrasal verb 'cover up'.
- Misplacing the stress: it's an 'OPEN COver', not an 'OPen COVer'.
Practice
Quiz
In which field is the term 'open cover' primarily used?
FAQ
Frequently Asked Questions
No, it is not a phrasal verb. It is a noun phrase, most commonly a technical term.
Only as a free combination of words (e.g., 'the open cover of the box'). As a fixed phrase with its technical meaning, it is confined to mathematical contexts.
An 'open cover' specifies that all sets in the covering collection are open sets. A 'cover' does not specify the nature of the sets.
In technical contexts, 'open covering' is a direct synonym. There is no simple, non-technical synonym for the mathematical concept.