lindelof space: meaning, definition, pronunciation and examples
C2/Very LowTechnical/Formal (Academic Mathematics, specifically Topology)
Quick answer
What does “lindelof space” mean?
A topological space in which every open cover has a countable subcover.
Audio
Pronunciation
Definition
Meaning and Definition
A topological space in which every open cover has a countable subcover.
A fundamental concept in topology, especially in the study of separation axioms, compactness, and covering properties. Named after the Finnish mathematician Ernst Lindelöf. A Lindelöf space is a weakening of compactness, replacing 'finite subcover' with 'countable subcover'.
Dialectal Variation
British vs American Usage
Differences
No difference in meaning or usage. Potential orthographic variation in handwriting/typing of the umlaut (ö) may occur, but the standard form 'Lindelöf' is universal in academic publications.
Connotations
None. Purely technical.
Frequency
Identically low frequency, confined exclusively to advanced mathematical discourse.
Grammar
How to Use “lindelof space” in a Sentence
[Topological Space X] + is + Lindelöf[Property/Theorem] + holds for + Lindelöf spacesProve/Show that + [Space] + is LindelöfVocabulary
Collocations
Examples
Examples of “lindelof space” in a Sentence
adjective
British English
- The Sorgenfrey plane is a classic example of a non-Lindelöf topological space.
American English
- A key step was proving the given manifold was, in fact, Lindelöf.
Usage
Meaning in Context
Business
Not used.
Academic
Exclusively used in advanced mathematics, particularly in topology and functional analysis papers, textbooks, and lectures.
Everyday
Never used.
Technical
The primary and only context. Used with precise, formal definitions and theorems.
Vocabulary
Synonyms of “lindelof space”
Weak
Vocabulary
Antonyms of “lindelof space”
Watch out
Common Mistakes When Using “lindelof space”
- Misspelling the name (Lindeloff, Lindelof without umlaut).
- Confusing with 'locally compact' or 'paracompact'.
- Incorrectly assuming all second-countable spaces are Lindelöf (true, but converse is false).
FAQ
Frequently Asked Questions
Yes. By definition, a compact space has a finite subcover for every open cover, and a finite set is countable. Therefore, compactness implies the Lindelöf property.
Yes, the real line ℝ is Lindelöf. This follows from it being second-countable. More directly, any open cover can be reduced to a countable subcover using the countability of the rationals.
A space is σ-compact if it is a countable union of compact subspaces. Every σ-compact space is Lindelöf, but the converse is not true. The Sorgenfrey line is Lindelöf but not σ-compact.
It is a crucial weak covering property that often serves as a sufficient condition for important theorems in topology and analysis (e.g., about continuity, convergence, or measure) when full compactness is not available. It bridges the gap between countable and arbitrary operations.
A topological space in which every open cover has a countable subcover.
Lindelof space is usually technical/formal (academic mathematics, specifically topology) in register.
Lindelof space: in British English it is pronounced /ˈlɪn.də.lɜːf speɪs/, and in American English it is pronounced /ˈlɪn.də.lɔːf speɪs/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think: 'Lindelöf lets you find a countable lift from an open cover.' The 'löf' can remind you of 'lift' a subcover.
Conceptual Metaphor
A space is 'Lindelöf' if, no matter how messily you blanket it (open cover), you can always find a countable collection of those blankets that still does the job.
Practice
Quiz
Which of the following statements about Lindelöf spaces is generally TRUE?