Reference: Cheeger, J., Colding, T., _On the structure of spaces with Ricci curvature bounded below., J. Differential Geometry, 45 (1997) 406--480.
I have a question regarding the following definition made in the above reference. Let $\dim$ denote the Hausdorff dimension. We say that $y$ is an $\ell$--dimensional point (where $\ell \geq 0$) if $$\lim_{r \to 0} \dim(B_r(y)) = \ell.$$
I am having a difficult time coming up with examples of $\ell$--dimensional points for $\ell > 0$. This leads me to ask the following questions:
Q1. Can someone provide a simple example of an $\ell$--dimensional point, where $\ell > 0$?
Q2. Suppose that $M$ is a smooth manifold and $y \in M$ is some point. Is it always true that $y$ is $0$--dimensional? In essence, do higher-dimensional points arise only in singular settings?