uniform ultrafilter (english) - meaning, definition, translation, pronunciation

Part of Speech

Noun

Phonetic Transcription

/ˈjuːnɪfɔrm ˈʌltrəˌfaɪltər/

Meaning and Usage

A uniform ultrafilter is a concept from set theory and mathematical logic, specifically in the study of filters and ultrafilters on a partially ordered set or topological space. It is a special type of ultrafilter that assures uniform convergence properties for a sequence of subsets of a given set. In essence, it is a mathematical tool used to analyze and define various properties of collections of sets or functions.

• Frequency of Use: The term is primarily used in academic and research contexts, particularly in mathematics, logic, and related fields. It is less common in everyday conversation.
• Context: The term is predominantly used in written context, such as academic papers, textbooks, and presentations.

Example Sentences

1. The research focused on demonstrating that a uniform ultrafilter exists on every non-empty set.

2. Исследование сосредоточилось на демонстрации того, что на каждом непустом множестве существует униформационный ультрафильтр.

3. Using a uniform ultrafilter, we can generalize the notion of convergence in topology.

4. Используя униформационный ультрафильтр, мы можем обобщить понятие сходимости в топологии.

5. The properties of a uniform ultrafilter are critical in understanding the structure of compact spaces.

6. Свойства униформационного ультрафильтра важны для понимания структуры компактных пространств.

Idiomatic Expressions

The term "uniform ultrafilter" does not appear commonly in idiomatic expressions due to its specialized mathematical nature. However, related expressions in mathematical contexts may occur.

Example Sentences

1. The concept of a limit depends heavily on the uniform ultrafilter in ensuring convergence.

2. Понятие предела сильно зависит от униформационного ультрафильтра, обеспечивающего сходимость.

3. When dealing with sequences, a uniform ultrafilter can provide a consistent framework for analysis.

4. При работе с последовательностями униформационный ультрафильтр может предоставить последовательную структуру для анализа.

5. The uniform ultrafilter played a crucial role in simplifying the proof of the main theorem.

6. Униформационный ультрафильтр сыграл ключевую роль в упрощении доказательства главной теоремы.

Etymology

The term "ultrafilter" originated in the field of topology and mathematical logic. It combines "ultra," meaning "beyond" or "exceeding," with "filter," which in mathematics refers to a class of subsets that captures certain closure properties under intersection and supersets. "Uniform" adds the dimension of consistency or sameness across its elements, pointing to its essential characteristics in set structures.

Synonyms and Antonyms

• Synonyms: None widely recognized due to the specificity of the term.
• Antonyms: Standard filter (as it does not have the same strict convergence properties as an ultrafilter).

Given the highly specialized nature of the term "uniform ultrafilter," the understanding is mostly confined to the domain of mathematical logic and set theory, making it less applicable in day-to-day language or idiomatic expressions.