closed homology group (english) - meaning, definition, translation, pronunciation

Part of Speech

Noun

Phonetic Transcription

/kləʊzd həˈmɒlədʒi ɡruːp/

Meaning and Usage

A "closed homology group" is a concept used in algebraic topology, a field of mathematics. It refers to a specific type of homology group that is associated with a topological space, typically used to study its properties through algebraic means. Homology groups provide a way to classify and measure the "holes" in a space. The "closed" aspect often pertains to the types of chains being considered, typically closed chains or cycles.

Closed homology groups are used primarily in written mathematical contexts, particularly in research papers and textbooks within the field of topology and related disciplines. The frequency of use can vary but is generally found in more advanced mathematical discussions.

Example Sentences

1. The closed homology group of a circle is isomorphic to the integers.

2. Translation: Le groupe d'homologie fermé d'un cercle est isomorphe aux entiers.

3. In proving the properties of closed homology groups, one must consider the structure of the underlying space.

4. Translation: En prouvant les propriétés des groupes d'homologie fermés, on doit prendre en compte la structure de l'espace sous-jacent.

5. Closed homology groups play a crucial role in the classification of manifolds.

6. Translation: Les groupes d'homologie fermés jouent un rôle crucial dans la classification des variétés.

Idiomatic Expressions

While "closed homology group" may not appear frequently in idiomatic expressions, it is embedded in mathematical jargon that can lead to specific phrases in this field. Here are other expressions using "homology" that may be encountered:

1. "Homology speaks volumes about the topological properties of the manifold."

2. Translation: L'homologie en dit long sur les propriétés topologiques de la variété.

3. "Clean homology can simplify our understanding of complex structures."

4. Translation: Une homologie propre peut simplifier notre compréhension des structures complexes.

5. "The strength of homology lies in its ability to reveal hidden connections."

6. Translation: La force de l'homologie réside dans sa capacité à révéler des connexions cachées.

Etymology

The term "homology" is derived from the Greek word "homologia" (ὁμολογία), meaning "agreement" or "relation." The prefix "closed" in mathematical terms typically indicates that a certain type of end behavior or condition is met, often related to boundary conditions in topological spaces.

Synonyms and Antonyms

Synonyms: - Closed chain group - Cycle group

Antonyms: - Open homology group - Non-closed chain group

This comprehensive information provides insight into the term "closed homology group," its mathematical context, and how it is used within the realm of topology.