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

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 $\mathbb R^n$.