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topological group: meaning, definition, pronunciation and examples

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UK/ˌtɒp.əˈlɒdʒ.ɪ.kəl ɡruːp/US/ˌtɑː.pəˈlɑː.dʒɪ.kəl ɡruːp/

Formal, academic, technical

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Quick answer

What does “topological group” mean?

A group endowed with a topology such that the group operations are continuous.

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Pronunciation

Definition

Meaning and Definition

A group endowed with a topology such that the group operations are continuous.

A set that is simultaneously a group (algebraic structure) and a topological space, where the algebraic operations of multiplication and inversion are continuous functions with respect to the given topology.

Dialectal Variation

British vs American Usage

Differences

No lexical differences; identical in both dialects.

Connotations

Exclusively denotes a technical mathematical concept.

Frequency

Used with identical rarity and exclusively within mathematics in both varieties.

Grammar

How to Use “topological group” in a Sentence

The set G, equipped with the product topology, forms a topological group.X is a topological group under componentwise addition.

Vocabulary

Collocations

strong
abelian topological groupcompact topological grouplocally compact topological groupHausdorff topological group
medium
structure of a topological groupstudy of topological groupscontinuous homomorphism of topological groups
weak
the group is topologicaltopological group theoryexample of a topological group

Examples

Examples of “topological group” in a Sentence

adjective

British English

  • The topological-group structure is essential for the proof.
  • We investigate topological-group properties like local compactness.

American English

  • The topological group structure is crucial for the proof.
  • We study topological group properties such as local compactness.

Usage

Meaning in Context

Business

Never used.

Academic

Core term in graduate-level mathematics, specifically in topology, abstract algebra, functional analysis, and representation theory.

Everyday

Never used.

Technical

Defined and used with precision in mathematical research papers, textbooks, and lectures.

Vocabulary

Synonyms of “topological group”

Strong

continuous group

Weak

topologized group

Vocabulary

Antonyms of “topological group”

group without topologydiscrete group (in specific contexts where discreteness contrasts with a richer topology)

Watch out

Common Mistakes When Using “topological group”

  • Confusing with 'topological space' (a more general concept).
  • Omitting the continuity requirement for the inverse operation.
  • Using it outside of a strictly mathematical context.

FAQ

Frequently Asked Questions

It is a compound noun phrase, always written as two words.

Yes. The set of real numbers R with the operation of ordinary addition and the standard Euclidean topology is a topological group.

A 'group' is a purely algebraic set with an operation. A 'topological group' adds a topology (a notion of 'closeness') and requires the group operation and inversion to be continuous with respect to that topology.

Only if they are studying advanced mathematics in English. It is not part of general English vocabulary.

Topological group is usually formal, academic, technical in register.

Topological group: in British English it is pronounced /ˌtɒp.əˈlɒdʒ.ɪ.kəl ɡruːp/, and in American English it is pronounced /ˌtɑː.pəˈlɑː.dʒɪ.kəl ɡruːp/. Tap the audio buttons above to hear it.

Learning

Memory Aids

Mnemonic

Think: A GROUP of friends who live on a weirdly shaped MAP (topology); visiting each other (group operation) must be a continuous journey.

Conceptual Metaphor

A machine (group) whose gears and levers move smoothly without jumps (topology).

Practice

Quiz

Fill in the gap
A is a set which is both an algebraic group and a topological space, with continuous group operations.
Multiple Choice

In which field is the term 'topological group' exclusively used?