I am reading Glen E. Bredon's Topology and Geometry and trying to solve the following problem in the book. But I can't understand what the blue g means. Could anyone explain please?
- Show that a second countable Hausdorff space $X$ with a functional structure $F$ is an $n$-manifold $\Leftrightarrow$ every point in $X$ has a neighborhood $U$ such that there are functions $f_{1},...,f_{n}\in F(U)$ such that: a real valued function $\color{blue}g$ on $U$ is in $F(U)$$\Leftrightarrow$there exists a smooth function $h(x_{1},...,x_{n})$ of $n$ real variables $\ni$ $g(p)=h(f_{1}(p),...,f_{n}(p))$ for all $p\in U$.
This means: $F(U)$ consists precisely of the functions that can be expressed as the composition $p\mapsto h(f_{1}(p),...,f_{n}(p))$ for some smooth $h$.
It's like describing the set $\mathbb{Q}$ of rational numbers by saying: $x\in\mathbb{Q}$ $\Leftrightarrow$ there exist integers $m,n$ such that $x=m/n$. Here, $x$ is a placeholder name we use to refer to a generic element of $\mathbb{Q}$. Same with $g$ in your question.
When you move past the definition toward the proof, $f_1,\dots, f_n$ will probably be the coordinate components of a chart map from $U$ into $\mathbb{R}^n$.