Discussion Overview
The discussion centers around the possibility of constructing a homeomorphism between R^m and a subset of R^n when m > n. Participants explore theoretical implications and specific cases, including the use of Brouwer's Invariance of Domain theorem to address the question.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant asserts that there does not exist a homeomorphism between R^m and R^n if m > n, questioning if a homeomorphism can exist between R^m and a subset of R^n.
- Another participant points out that any subspace of R^n is R^k for k < n < m, reinforcing the original claim about homeomorphisms.
- A different participant proposes considering subsets of R^n that are not the whole R^k, such as ill-behaved sets like space-filling lines, suggesting a potential exception to the general rule.
- One participant discusses Brouwer's Invariance of Domain theorem as a tool to prove the non-existence of a homeomorphism, outlining a specific argument involving the mapping of R^m to a subset S of R^n and the implications of this mapping.
- A later reply expresses gratitude for the explanation provided regarding the use of Brouwer's theorem, indicating that the proof was appreciated.
- Another participant suggests checking a related thread for additional context or information.
Areas of Agreement / Disagreement
Participants generally agree on the foundational claim regarding homeomorphisms between R^m and R^n when m > n, but there is contention regarding the possibility of homeomorphisms involving subsets of R^n, particularly with ill-behaved sets. The discussion remains unresolved with multiple viewpoints presented.
Contextual Notes
The discussion involves assumptions about the nature of subsets and the application of Brouwer's theorem, which may not cover all cases or definitions of homeomorphism.