Discussion Overview
The discussion revolves around identifying a specific term for a subset of real space that is not a manifold with boundary. Participants explore the properties of such sets, particularly focusing on conditions where local homeomorphism to Euclidean spaces or half-spaces does not hold.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant inquires about a name for a subset S of \(\mathbb{R}^n\) that lacks local homeomorphic properties to either \(\mathbb{R}^m\) or the half-space \(\mathbb{H}^m\) for any integer \(m \geq 0\).
- Another participant suggests that the complement of a dense set might be relevant to the original question.
- A participant refines their question, indicating that while the complement of a dense set could be a correct answer for the case where \(m = n\), it does not hold for general \(m\). They provide an example involving the graph of a nowhere continuous function.
- Further reflection leads to the conclusion that the half-space condition may be unnecessary, as local isomorphism to a half-space implies isomorphism to Euclidean space nearby.
- Participants acknowledge a misunderstanding regarding the condition \(m \neq n\) and express that it was not included in the original formulation.
- One participant maintains that the complement of a dense set is still a necessary condition, but notes that not every complement of a dense set is suitable, citing the example of \(\{0\}\).
Areas of Agreement / Disagreement
Participants express differing views on the correct characterization of the set in question, with no consensus reached on a specific term or definition. The discussion remains unresolved regarding the properties and naming of such sets.
Contextual Notes
Participants note limitations in their definitions and examples, particularly regarding the necessity of the half-space condition and the implications of the complement of a dense set. The discussion reflects ongoing exploration rather than established conclusions.
