Podcast
Questions and Answers
Soham's ball is a topological manifold.
Soham's ball is a topological manifold.
False
Soham's ball can be subdivided into a finite number of simplices.
Soham's ball can be subdivided into a finite number of simplices.
False
Soham's ball is a compact, connected, and simply-connected space.
Soham's ball is a compact, connected, and simply-connected space.
True
Soham's ball is used as an example to illustrate the limitations of certain topological properties.
Soham's ball is used as an example to illustrate the limitations of certain topological properties.
Signup and view all the answers
Soham's ball is a two-dimensional topological space.
Soham's ball is a two-dimensional topological space.
Signup and view all the answers
Study Notes
Soham's Ball
Soham's ball is a concept in mathematics, specifically in the field of topology.
Definition
Soham's ball is a topological space that is defined as the quotient space of a 3-ball (a three-dimensional ball) by identifying the northern hemisphere with the southern hemisphere in a certain way.
Properties
- Soham's ball is a non-manifold, meaning it is not a topological manifold.
- It is a compact, connected, and simply-connected space.
- Soham's ball is not a triangulable space, meaning it cannot be subdivided into a finite number of simplices (i.e., points, lines, triangles, etc.).
Significance
Soham's ball is an important example in topology because it shows that not all compact, connected, and simply-connected spaces are manifolds. It is often used as a counterexample to illustrate the limitations of certain topological properties.
Soham's Ball
- A topological space in mathematics, specifically in the field of topology.
Definition
- Soham's ball is a quotient space of a 3-ball (a three-dimensional ball).
- The northern hemisphere is identified with the southern hemisphere in a certain way.
Properties
- Non-manifold, meaning it is not a topological manifold.
- Compact, connected, and simply-connected space.
- Not a triangulable space, meaning it cannot be subdivided into a finite number of simplices (e.g., points, lines, triangles, etc.).
Significance
- Important example in topology, showing that not all compact, connected, and simply-connected spaces are manifolds.
- Often used as a counterexample to illustrate the limitations of certain topological properties.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Learn about Soham's ball, a topological space with unique properties, including being non-manifold, compact, connected, and simply-connected.
