Is $A=\{ x \in \mathbb{R}^2 : ||x|| \leq 1 \}$ a 1d or 2d manifold

Question

Suppose the set $A=\{ x \in \mathbb{R}^2 : ||x|| \leq 1 \} $.
Can A be a 1d or 2d or n-dimentional manifold ?

My thought is that we can write A as $A=A_1\cup A_2 = \{ x \in \mathbb{R}^2 : ||x|| < 1 \} \cup \{ x \in \mathbb{R}^2 : ||x||=1 \}$ and cover A with the charts:

For $A_1$: $f:A_1 \rightarrow \mathbb{R} \quad f(x,y)=\frac{x}{1-||(x,y)||}$
For $A_2$: The stereographic projection $f_N:A_1-\{(0,1)\} \rightarrow \mathbb{R}$ and $f_S:A_1-\{(0,-1)\} \rightarrow \mathbb{R}$

So it is a 1d manifold but it can't be a 2d manifold because there can not be an open set in $\mathbb{R}^2$ homeomorphic to a subset of $A_2$

Answer 1

You might want to read this about manifolds with boundaries.