Topologie (with Heinz Hopf)
E173182
"Topologie" is a foundational 1935 textbook on general topology co-authored by Pavel Alexandrov and Heinz Hopf that helped shape the modern development of the field.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Topologie (with Heinz Hopf) canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1509458 — 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: Topologie (with Heinz Hopf) Context triple: [Pavel Alexandrov, notableWork, Topologie (with Heinz Hopf)]
-
A.
Theorie der Transformationsgruppen
Theorie der Transformationsgruppen is Sophus Lie’s foundational multi-volume work that established the theory of continuous transformation groups, now known as Lie groups, and their applications to differential equations and geometry.
-
B.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
-
C.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
-
D.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
E.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
- 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: Topologie (with Heinz Hopf) Target entity description: "Topologie" is a foundational 1935 textbook on general topology co-authored by Pavel Alexandrov and Heinz Hopf that helped shape the modern development of the field.
-
A.
Theorie der Transformationsgruppen
Theorie der Transformationsgruppen is Sophus Lie’s foundational multi-volume work that established the theory of continuous transformation groups, now known as Lie groups, and their applications to differential equations and geometry.
-
B.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
-
C.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
-
D.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
E.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
- F. None of above. chosen
Statements (30)
| Predicate | Object |
|---|---|
| instanceOf |
mathematics textbook
ⓘ
nonfiction book ⓘ topology textbook ⓘ |
| author |
Heinz Hopf
ⓘ
Pavel Alexandrov ⓘ |
| coAuthor |
Heinz Hopf
ⓘ
Pavel Alexandrov ⓘ |
| contributedTo | axiomatic development of topology ⓘ |
| countryOfPublication | Germany ⓘ |
| describedAs | foundational textbook on general topology ⓘ |
| era | 20th-century mathematics literature ⓘ |
| field |
general topology
ⓘ
topology ⓘ |
| hasAuthorRole |
Heinz Hopf
ⓘ
Pavel Alexandrov ⓘ |
| influenced |
20th-century topology
ⓘ
modern general topology ⓘ |
| language | German ⓘ |
| publicationYear | 1935 ⓘ |
| topic |
compactness
ⓘ
connectedness ⓘ continuous mappings ⓘ fundamental concepts of general topology ⓘ metric spaces ⓘ product spaces ⓘ quotient spaces ⓘ separation axioms ⓘ topological spaces ⓘ |
| usedAs |
graduate-level textbook
ⓘ
reference work in topology ⓘ |
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: Topologie (with Heinz Hopf) Description of subject: "Topologie" is a foundational 1935 textbook on general topology co-authored by Pavel Alexandrov and Heinz Hopf that helped shape the modern development of the field.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.