The discussion centers on whether the coordinate axes in the plane, defined as the set $$ D=\{ (x,0) \in \mathbb{R}^2 : x \in \mathbb{R}\} \cup \{ (0,y) \in \mathbb{R}^2 : y \in \mathbb{R} \}$$, can be classified as a 1-dimensional or 2-dimensional differentiable manifold. The scope includes theoretical considerations of manifold properties and topology.
Participants generally disagree on the classification of the set D as a manifold. While some argue it is not a manifold due to the properties at the origin, others suggest that removing the origin allows it to be considered a topological manifold.
Limitations include the dependence on the definitions of manifold and topological space, as well as the unresolved nature of the implications of removing the origin on the manifold structure.
If you remove the origin, I believe it is a topological manifold.WWGD said:And removal of a single point, the origin, would disconnect it into 4 components, unlike any surface or 1-manifold. It's clearly not a differentiable ( if it was a manifold ), as its tangent space is not defined at the origin. It's not even a manifold with boundary, as no neighborhood of the origin is homeomorphic to a (subspace) neighborhood of the upper half plane .
Indeed, 4 lines, each a 1-manifold, globally homeomorphic to the Reals. A manifold with 4 connected components.jbergman said:If you remove the origin, I believe it is a topological manifold.