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.
A differentiable manifold is a mathematical concept used to describe spaces that are locally similar to Euclidean spaces, but may have more complex global structures. It is a generalization of the concept of a smooth curve or surface in traditional geometry.
This refers to the dimensionality of the manifold. A 1-dimensional differentiable manifold is a curve, while a 2-dimensional differentiable manifold is a surface. In general, a d-dimensional differentiable manifold is a space that locally resembles d-dimensional Euclidean space.
We can determine if the coordinate axes are a differentiable manifold by checking if they satisfy the definition of a differentiable manifold. This includes being locally homeomorphic to Euclidean space, having a smooth structure, and being Hausdorff and second-countable.
Knowing if the coordinate axes are a differentiable manifold is important in many areas of science and mathematics, such as differential geometry, topology, and physics. It allows us to study and understand the properties of these spaces and apply them to real-world problems.
Differentiable manifolds have many applications in various fields, including computer graphics, robotics, image processing, and machine learning. They are also used in physics to describe the geometry of spacetime in general relativity and in the study of dynamical systems and chaos theory.