CW complexes
E911364
CW complexes are topological spaces built by inductively attaching cells of increasing dimension, providing a flexible and combinatorially tractable framework for algebraic topology.
All labels observed (1)
| Label | Occurrences |
|---|---|
| CW complexes canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11219611 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: CW complexes Context triple: [Characteristic Classes, hasSubject, CW complexes]
-
A.
Eilenberg–MacLane spaces
Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
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B.
Eilenberg–Steenrod axioms
The Eilenberg–Steenrod axioms are a foundational set of conditions that formally characterize homology theories in algebraic topology.
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C.
Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
-
D.
Whitney approximation theorem
The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
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E.
Alexander–Spanier cohomology
Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: CW complexes Target entity description: CW complexes are topological spaces built by inductively attaching cells of increasing dimension, providing a flexible and combinatorially tractable framework for algebraic topology.
-
A.
Eilenberg–MacLane spaces
Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
-
B.
Eilenberg–Steenrod axioms
The Eilenberg–Steenrod axioms are a foundational set of conditions that formally characterize homology theories in algebraic topology.
-
C.
Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
-
D.
Whitney approximation theorem
The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
-
E.
Alexander–Spanier cohomology
Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical structure
ⓘ
topological space ⓘ |
| exampleOf | cell complex ⓘ |
| field | algebraic topology NERFINISHED ⓘ |
| generalizes |
finite cell complexes
ⓘ
simplicial complexes ⓘ |
| hasProperty |
0-skeleton is a discrete set of points
ⓘ
Hausdorff NERFINISHED ⓘ admits CW-structure on many familiar spaces ⓘ admits cellular approximation of maps ⓘ attaching maps determine the CW structure ⓘ built from open cells ⓘ built inductively by attaching cells ⓘ cellular chain complex computes homology ⓘ cellular cochain complex computes cohomology ⓘ closed under homotopy equivalence up to CW-approximation ⓘ closure-finite condition on cells ⓘ combinatorially tractable ⓘ constructed by attaching n-dimensional cells ⓘ countable CW complexes have countably many cells ⓘ each cell attached via continuous map from boundary sphere ⓘ every CW complex is weakly homotopy equivalent to a simplicial complex under mild conditions ⓘ finite CW complexes have finitely many cells ⓘ flexible for constructions in topology ⓘ good for computing homotopy type ⓘ has a filtration by skeleta ⓘ locally finite in many applications ⓘ n-skeleton is obtained by attaching n-cells to (n-1)-skeleton ⓘ subcomplexes are unions of cells ⓘ supports Whitehead theorem for CW complexes ⓘ supports cellular approximation theorem ⓘ supports obstruction theory ⓘ weak topology condition on closures of cells ⓘ weak topology with respect to its cells ⓘ well-behaved with respect to homotopy ⓘ |
| introducedBy | J. H. C. Whitehead NERFINISHED ⓘ |
| introducedIn | 20th century ⓘ |
| nameExpandsTo | closure-finite weak topology complex ⓘ |
| typicalExample |
Eilenberg–MacLane spaces
NERFINISHED
ⓘ
Grassmannians with CW structure ⓘ classifying spaces of groups ⓘ projective spaces with cell structure ⓘ spheres with standard cell decomposition ⓘ |
| usedIn |
cellular homology
ⓘ
cohomology theory ⓘ homology theory ⓘ homotopy groups of spheres ⓘ homotopy theory ⓘ spectral sequences ⓘ stable homotopy theory ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: CW complexes Description of subject: CW complexes are topological spaces built by inductively attaching cells of increasing dimension, providing a flexible and combinatorially tractable framework for algebraic topology.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.