locally compact topology (english) - meaning, definition, translation, pronunciation

Part of Speech

Noun Phrase

Phonetic Transcription

/ˈloʊkəli kɑːmˈpækt ˈtɑːpəˌlɑːdʒi/

Meaning and Usage

"Locally compact topology" is a term used in mathematics, specifically in the field of topology. It refers to a topological space that, at a local level, resembles compact spaces. More technically, a topological space is said to be locally compact if every point has a neighborhood that is contained in a compact subset.

Usage in English

This term is primarily used in academic and mathematical writings, making it more common in written context than in oral speech. It is often encountered in higher-level mathematics courses and research papers.

Example Sentences

1. In order to apply theorems from analysis, it is crucial to understand locally compact topology.

2. Для применения теорем из анализа важно понимать локально компактную топологию.

3. Many properties of locally compact topologies parallel those found in compact spaces.

4. Многие свойства локально компактных топологий аналогичны тем, что наблюдаются в компактных пространствах.

5. The study of locally compact topology has significant implications in functional analysis.

6. Изучение локально компактной топологии имеет важные последствия в функциональном анализе.

Idiomatic Expressions

While "locally compact topology" does not appear in idiomatic expressions, it is foundational in discussing concepts in topology that may have idiomatic references in relation to phenomena like compactness or continuity.

Example Sentences with Related Idiomatic Expressions

1. The compact sets in this space provide a solid foundation for understanding continuity.

2. Компактные множества в этом пространстве дают прочную основу для понимания непрерывности.

3. When studying topology, one must grasp the notion of convergence to avoid losing track of the big picture.

4. При изучении топологии необходимо понимать понятие сходимости, чтобы не потерять из виду общую картину.

5. In analysis, compactness is key to unlocking many important results.

6. В анализе компактность является ключом к раскрытию многих важных результатов.

Etymology

The term "locally compact topology" can be broken down into its components: - "Locally" derives from the Latin word "localis," referring to a specific area or vicinity. - "Compact" comes from the Latin "compactus," meaning "pressed together." - "Topology" stems from the Greek "topos," meaning "place," and the suffix "-logy" from the Greek "logia," meaning "study of."

Synonyms and Antonyms

Synonyms

• Locally compact space
• Compactly generated space

Antonyms

• Non-compact space
• Non-locally compact space

This comprehensive overview offers insights into the term "locally compact topology," covering various aspects ranging from meaning and usage to etymology and related terms.