locally euclidean space: meaning, definition, pronunciation and examples
C2+ (Academic/Technical)Formal Academic/Technical
Quick answer
What does “locally euclidean space” mean?
A topological space where every point has a neighbourhood that is homeomorphic to an open subset of Euclidean space of some fixed dimension n.
Audio
Pronunciation
Definition
Meaning and Definition
A topological space where every point has a neighbourhood that is homeomorphic to an open subset of Euclidean space of some fixed dimension n.
In topology and geometry, it is a key concept for defining manifolds. It describes spaces that, when 'zoomed in' sufficiently around any point, look like ordinary flat (Euclidean) space, even if the overall space is curved or complicated.
Dialectal Variation
British vs American Usage
Differences
No lexical or spelling differences. The term is identical in both dialects.
Connotations
Exclusively technical, with no cultural or informal connotations.
Frequency
Extremely rare outside of advanced mathematics, topology, and theoretical physics. Frequency is identical across dialects in relevant academic fields.
Grammar
How to Use “locally euclidean space” in a Sentence
[Space] is a locally euclidean space.The definition of a manifold requires a locally euclidean space.One can prove that [object] is locally euclidean.Vocabulary
Collocations
Examples
Examples of “locally euclidean space” in a Sentence
adjective
British English
- The locally euclidean property is fundamental.
- We need a locally euclidean structure.
American English
- The locally Euclidean property is fundamental.
- We need a locally Euclidean structure.
Usage
Meaning in Context
Business
Not applicable.
Academic
Core term in advanced mathematics courses and research papers on topology, differential geometry, and manifold theory.
Everyday
Never used.
Technical
The primary context. Used to define the local structure of manifolds in pure mathematics, mathematical physics, and some areas of computer graphics/vision.
Vocabulary
Synonyms of “locally euclidean space”
Neutral
Vocabulary
Antonyms of “locally euclidean space”
Watch out
Common Mistakes When Using “locally euclidean space”
- Omitting 'locally' and just saying 'euclidean space', which has a very different (global) meaning.
- Confusing it with a 'manifold', which typically adds conditions like being Hausdorff and second-countable.
- Using it in non-mathematical contexts.
FAQ
Frequently Asked Questions
Not quite. 'Manifold' (specifically a topological manifold) usually requires the space to also be Hausdorff and second-countable. A locally euclidean space is the core local property needed for a manifold.
Typically, in standard definitions, the dimension n is fixed for the entire space. If the dimension varies from point to point, it is not usually called a locally euclidean space in the sense of defining a manifold.
Almost exclusively in university-level mathematics courses on topology (e.g., point-set topology, differential topology) and in research literature in geometry, topology, and theoretical physics (e.g., general relativity).
The circle is a 1-dimensional locally euclidean space. Any small arc of the circle is topologically equivalent to an open interval on the real line (which is Euclidean 1-space). Similarly, the surface of a torus (doughnut) is a 2-dimensional locally euclidean space.
Locally euclidean space is usually formal academic/technical in register.
Locally euclidean space: in British English it is pronounced /ˌləʊkəli juːˈklɪdiən speɪs/, and in American English it is pronounced /ˌloʊkəli juˈklɪdiən speɪs/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of the surface of the Earth: globally it's a sphere, but LOCALLY, in your immediate neighbourhood, it appears flat or EUCLIDEAN. Your neighbourhood is like a small, flat map of the area—a LOCALLY EUCLIDEAN SPACE.
Conceptual Metaphor
ZOOMING IN FOR SIMPLICITY: A complex, potentially curved object is understood by imagining that if you look very closely at any single point, it resembles simple, familiar, flat geometry.
Practice
Quiz
What does it mean for a space to be 'locally euclidean'?