The discussion revolves around the question of why a branched line in R2 does not qualify as a topological manifold. Participants explore the definitions and properties of topological manifolds, particularly focusing on the existence of charts at points of branching.
Participants express uncertainty and seek clarification on the definitions and properties involved. There is no consensus reached regarding the construction of a chart or the implications of connectedness in this context.
Limitations include the participants' varying levels of understanding of topological concepts and the definitions of open sets within the standard topology on R2.
This discussion may be useful for individuals studying topology, particularly those interested in the properties of manifolds and the implications of branching in geometric contexts.
How do you suggest such a chart is constructed?CSteiner said:but I don't understand why not.
Why not? Sorry if this seems like a stupid question, but I'm only just now learning (and trying to understand) the definitions of these things. Open means that it belongs the standard topology on R2, right? Why doesn't it?Orodruin said:But it is not a mapping from an open subset of R^2.
CSteiner said:Is there a topologist out there that wants to explain why exactly a branched line in R2 is not not a topological manifold? I know it's because there doesn't exist a chart at the point of branching, but I don't understand why not. I'm just starting to self study this, so go easy on me :).
Not rigorously.micromass said:Do you know the concept of "connected"?
Will do, thanks for pointing me in the right direction.micromass said:You should learn about that then. Learn about "cut points" too. With that you can prove your OP.