# Are the coordinate axes a 1d- or 2d-differentiable manifold?

• I
• Delong66
In summary, the set D, which is a subset of R^2 with subspace topology, is not a manifold. It does not satisfy the requirement of having a neighborhood homeomorphic to an open subset of Euclidean space for every point. Additionally, removing the origin would result in 4 disconnected components, making it different from any surface or 1-manifold. However, if the origin is removed, it can be considered a topological manifold with 4 connected components, each being a 1-manifold and globally homeomorphic to the Reals.

#### Delong66

Suppose $$D=\{ (x,0) \in \mathbb{R}^2 : x \in \mathbb{R}\} \cup \{ (0,y) \in \mathbb{R}^2 : y \in \mathbb{R} \}$$ is a subset of $$\mathbb{R}^2$$ with subspace topology. Can this be a 1d or 2d manifold?
Thank you!

The set D is not a manifold. Every point in a manifold must have a neighbourhood that is homeomorphic to an open subset of a Euclidean space. The point (0,0) in set D has no such neighbourhood, as any open set containing (0,0) has an intersection of the two lines in it, and neither 1D nor 2D Euclidean space has an open subset consisting of such an intersection.

Orodruin
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 .

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 .
If you remove the origin, I believe it is a topological manifold.

jbergman said:
If you remove the origin, I believe it is a topological manifold.
Indeed, 4 lines, each a 1-manifold, globally homeomorphic to the Reals. A manifold with 4 connected components.