John Lee : Cubical charts and cube in $\mathbb{R}^n$

Question

What are the definitions of

1. cubical chart for a smooth manifold
2. cube in $\mathbb{R}^n$

I am reading John Lee's Introduction to Smooth Manifolds 2nd edition, and the author seems to use these mathematical objects often but I'm unable to get a precise definition for them.

I have an interpretation of those definitions as

1. cube in $\mathbb{R}^n$ is a product of connected open sets in $\mathbb{R} \times \mathbb{R} \ldots \times \mathbb{R}$
2. cubical chart is $(U,\varphi)$ where $U$ is open in the manifold and $\varphi(U)$ is a cube in $\mathbb{R}^n$

Are these interpretations correct?

Answer 1

The index is your friend! Cube is defined on page 649, coordinate cube on page 4, and smooth coordinate cube on page 15. (Note that, as I wrote in the preface, most readers should read, or at least skim, the appendices before the rest of the book.) "Cubical" is the adjective form of "cube," so a cubical chart is just a chart whose domain is a coordinate cube, or equivalently whose image is an open cube in $\mathbb R^n$.

So your interpretations are close, but not exactly right. More precisely,

1. An open cube in $\mathbb R^n$ is a product of bounded open intervals that all have the same length.
2. A cubical chart is a coordinate chart $(U,\varphi)$, where $U$ is open in the manifold and $\varphi(U)$ is an open cube in $\mathbb R^n$.