locally compact space: meaning, definition, pronunciation and examples
C2+Academic/Technical (Mathematics, specifically Topology and Functional Analysis)
Quick answer
What does “locally compact space” mean?
A topological space in which every point has a compact neighbourhood.
Audio
Pronunciation
Definition
Meaning and Definition
A topological space in which every point has a compact neighbourhood.
In topology and analysis, a topological space where every point possesses a neighbourhood that is compact (i.e., every open cover has a finite subcover). This is a fundamental property that lies between compactness and the Hausdorff property, crucial for results like Tychonoff's theorem for locally compact spaces and the construction of one-point compactifications.
Dialectal Variation
British vs American Usage
Differences
No significant difference in meaning or usage. Orthography follows standard regional conventions (e.g., 'neighbourhood' vs. 'neighborhood') in explanatory prose, but the formal term itself is invariant.
Connotations
None beyond its precise mathematical definition.
Frequency
Exclusively used within advanced mathematics. Frequency is identical in both academic communities.
Grammar
How to Use “locally compact space” in a Sentence
[The topological space X] is locally compact.A locally compact [space/group/topological vector space].The property of [being/having] locally compact.Vocabulary
Collocations
Examples
Examples of “locally compact space” in a Sentence
adjective
British English
- The locally compact case is simpler for integration theory.
American English
- A locally compact Hausdorff space has a well-defined one-point compactification.
Usage
Meaning in Context
Business
Not used.
Academic
Central term in topology, functional analysis (e.g., in the study of C*-algebras via the Gelfand representation), harmonic analysis (on locally compact groups), and measure theory.
Everyday
Never used.
Technical
Used identically to academic context; the primary domain of use.
Vocabulary
Synonyms of “locally compact space”
Weak
Vocabulary
Antonyms of “locally compact space”
Watch out
Common Mistakes When Using “locally compact space”
- Using 'locally' to mean 'sometimes' or 'in some areas' instead of 'in the neighbourhood of every point'.
- Confusing it with 'compact' alone. A space can be compact (globally) without being locally compact (e.g., some non-Hausdorff spaces), and vice-versa (e.g., the real line).
- Attempting to use it in non-mathematical contexts.
FAQ
Frequently Asked Questions
Yes. In a compact space, the entire space is a compact neighbourhood of every point. However, the converse is not true (e.g., ℝ is locally compact but not compact).
Local compactness is often studied in conjunction with the Hausdorff (T₂) separation axiom. A locally compact Hausdorff space is completely regular (Tychonoff). Many important results, like the existence of the one-point compactification, require the Hausdorff condition.
Yes, an infinite-dimensional Hilbert space in its norm topology is not locally compact. Also, the rational numbers ℚ with the subspace topology from ℝ are not locally compact, as no neighbourhood of a point in ℚ has compact closure in ℚ.
It is crucial for extending measures (via the Riesz–Markov–Kakutani representation theorem), for harmonic analysis on groups (locally compact groups), and for non-commutative geometry where it generalises to locally compact quantum groups. It ensures the existence of 'enough' functions with compact support.
Locally compact space is usually academic/technical (mathematics, specifically topology and functional analysis) in register.
Locally compact space: in British English it is pronounced /ˌləʊ.kə.li ˈkɒm.pækt ˈspeɪs/, and in American English it is pronounced /ˌloʊ.kə.li kəmˈpækt ˈspeɪs/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think: 'Local news covers a compact area.' Similarly, in a locally compact space, around every point (locally) you can find a 'news area' (neighbourhood) that is itself compact (finite cover).
Conceptual Metaphor
MATHEMATICAL PROPERTY IS A LOCATION: 'Local' implies looking in the immediate vicinity of a point. COMPACTNESS IS FINITENESS/TIDINESS.
Practice
Quiz
Which of the following is a necessary and sufficient condition for a Hausdorff space to be locally compact?