Definition of coordinate ball

Question

I am trying to read some of smooth manifolds by Lee.

He defines a topological manifold of dimension $n$ as a topological space $M$ such that:

1. The topology is secound countable and hausdorff
2. For each point there is a nbr (open subset containing it) $U$ which is homemorphic to $U'$, an open subset of $R^n$

He then defines, given such $M$, a chart on $M$ to be a pair $(U,\phi)$, with $U$ an open subset of $M$ and $\phi$ a homemorphism into an open subset of $R^n$.

He then gives a confusing definition of a coordinate ball, it is a set $U$ such that $\phi(U)$ is an open ball.

It's not clear if we defined a coordinate ball as (1): any subset such that there exists such a $\phi$, or (2) with respect to a specific $\phi$.

This then gets more confusing as the first lemma is:

Every topological manifold has a countable basis of precompact coordinate balls.

(2) doesn't really make sense here, unless he implicitly means that given a manifold we consider the charts given by each point (that open nbr we assumed it had). This would be especially annoying since it disallows us to use easy reductions such that the open subset of $R^n$ is wlog all of $R^n$ (via taking a different nbr if needed).

So is (1) the correct definition?

Thanks

Answer 1

In the version of Lee that I can get on the internets (my copy of Lee is at the office), the definition given is contained a paragraph which reads

Suppose $M$ is locally Euclidean of dimension $n$. If $U\subseteq M$ is an open subset that is homeomorphic to an open subset of $\mathbb{R}^n$, then $U$ is called a coordinate domain, and any homeomorphism $\varphi$ from $U$ to an open subset of $\mathbb{R}^n$ is called a coordinate map. The pair $(U,\varphi)$ is called a coordinate chart (or just a chart) for $M$. A coordinate domain that is homeomorphic to a ball in $\mathbb{R}^n$ is called a coordinate ball. (When $M$ has dimension 2, we sometimes use the term coordinate disk.) The preceding lemma shows that every point in a locally Euclidean space is contained in a coordinate ball. If $p \in M$ and $U$ is a coordinate domain containing $p$, we say that $U$ is a coordinate neighborhood or Euclidean neighborhood of $p$.

Hence a coordinate ball is an open set $U$ such that there exists some coordinate chart (i.e. a homeomorphism) $\varphi$ such that $\varphi(U)$ is a ball in $\mathbb{R}^n$. This seems to be your Definition (1). Essentially, the idea is that coordinate domain is a subsets $U$ of $M$ such that we can impose coordinates on $U$ via some chart (any chart). A coordinate ball is a coordinate domain homeomorphic to a ball.