completely regular space: meaning, definition, pronunciation and examples
Very Low (Technical/Specialist)Formal Academic / Technical Mathematics
Quick answer
What does “completely regular space” mean?
A topological space where for any closed set and a point not in it, there exist disjoint open sets containing them, respectively.
Audio
Pronunciation
Definition
Meaning and Definition
A topological space where for any closed set and a point not in it, there exist disjoint open sets containing them, respectively.
In topology, a completely regular space (also called a Tychonoff space) satisfies a stronger separation axiom than a regular space; every closed set and point not in it can be functionally separated by a continuous real-valued function.
Dialectal Variation
British vs American Usage
Differences
No difference in meaning. The term is used identically in both mathematical communities.
Connotations
None beyond its strict mathematical definition.
Frequency
Extremely rare outside advanced mathematics textbooks and research papers.
Grammar
How to Use “completely regular space” in a Sentence
The space X is completely regular.A completely regular Hausdorff space.Prove that this space is completely regular.Vocabulary
Collocations
Examples
Examples of “completely regular space” in a Sentence
adjective
British English
- The proof relies on the space being completely regular.
- We assume a completely regular topological space.
American English
- They studied the properties of a completely regular space.
- A compact Hausdorff space is completely regular.
Usage
Meaning in Context
Business
Never used.
Academic
Exclusively used in advanced mathematics, specifically in topology and functional analysis courses and literature.
Everyday
Never used.
Technical
The primary and only context. Used in mathematical proofs, definitions, and theorems.
Vocabulary
Synonyms of “completely regular space”
Neutral
Vocabulary
Antonyms of “completely regular space”
Watch out
Common Mistakes When Using “completely regular space”
- Using it in a non-mathematical context.
- Confusing it with 'regular space' (a weaker condition).
- Omitting the 'completely'.
- Misspelling 'Tychonoff'.
FAQ
Frequently Asked Questions
A regular space can separate a point and a closed set with disjoint open sets. A completely regular space can do this via a continuous real-valued function (a stronger condition), implying it can also be done with open sets.
Not necessarily by definition, but it is almost always assumed to be T1 (points are closed), which makes it Hausdorff. The term 'Tychonoff space' specifically means a completely regular T1 space, which is Hausdorff.
Only in university-level courses or texts on general topology, functional analysis, or related fields of pure mathematics.
All metric spaces are normal, hence completely regular. The real number line with the standard topology is a classic example.
A topological space where for any closed set and a point not in it, there exist disjoint open sets containing them, respectively.
Completely regular space is usually formal academic / technical mathematics in register.
Completely regular space: in British English it is pronounced /kəmˌpliːtli ˈreɡjʊlə speɪs/, and in American English it is pronounced /kəmˌpliːtli ˈreɡjəlɚ speɪs/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think: 'Completely Regular' = Can 'Regularly' use a Continuous function to 'Completely' separate a point from a closed set.
Conceptual Metaphor
None; it is a purely abstract, logical property.
Practice
Quiz
Which of the following is a key property of a completely regular space?