The set D, defined as $$ D=\{ (x,0) \in \mathbb{R}^2 : x \in \mathbb{R}\} \cup \{ (0,y) \in \mathbb{R}^2 : y \in \mathbb{R} \}$$, is not a differentiable manifold due to the lack of a neighborhood around the point (0,0) that is homeomorphic to an open subset of Euclidean space. The intersection of the two lines at the origin prevents it from being classified as either a 1D or 2D manifold. Furthermore, removing the origin disconnects the set into four components, reinforcing its non-manifold status. However, if the origin is removed, D can be considered a topological manifold with four connected components, each homeomorphic to the real numbers.
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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.