Dimension of the real plane without origin Is there an intuitive way to deduce the dimension of $\mathbb{R}^2 \setminus \{(0,0)\}$ (real plane without origin), considered as a manifold?
The space $\mathbb{R}^2 \setminus \{(0,0)\}$ is homeomorphic to $\mathbb{R}^2$.  This is analogous to the way that a line is homeomorphic to a circle that is missing one point.  (See &quot;one-point compactification for details.&quot;)  Since $\mathbb{R}^2$ is a Euclidean plane with dimension $2$, your manifold has two dimensions.
The space $\mathbb{R}^2 \setminus \{(0,0)\}$ is homeomorphic to $\mathbb{R}^2$.  This is analogous to the way that a line is homeomorphic to a circle that is missing one point.  (See &quot;one-point compactification for details.&quot;)  Since $\mathbb{R}^2$ is a Euclidean plane with dimension $2$, your manifold has two dimensions.