# 0708-1300/Errata to Bredon's Book

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Problem 1, p. 71.

There is a counterexample to the inverse implication in Problem 1, p. 71.

Let $X=\mathbb{R}$ be endowed with the ordinary topology (thus, it is Hausdorff and second countable). Let $U$ be an arbitrary connected open set in $X$ (that is, an interval). Let $F_X(U)$ consists of all functions identically equal to constant. If $U$ is an arbitrary open set, then by theorem on structure of open sets in $\mathbb{R}$ it is a union of countably many open intervals. We define $F_X(U)$ to be the set of all real-valued functions which are constant on open intervals forming $U$. The family $F=\{F_X(U):U\mbox{ is open in }X\}$ forms a functional structure, as one can check. Furthermore, it satisfies the hypothesis of the theorem: every point $x \in X$ has a neighborhood (we take an open interval containing $x$) such that there exists a function $f \in F_X(U)$ (we define it to be identically equal to $1$) such that a function $g:U \to \mathbb{R}$ is in $F_X(U)$ (it is identically equal to a constant by our definition) if and only if there exists a smooth function $h$ such that $g=h \circ f$ (if $g$ is given, then we define $h(x)=g$ for all $x$, if $f$ is given, then we take arbitrary smooth $h:\mathbb{R} \to \mathbb{R}$, since $h \circ f$ is identically equal to constant and, thus, is in $F_X(U)$). Clearly, $(X,F_X)$ is not a smooth manifold. Even taking $X$ as any $T_2$ second countable topological space with the functional structure of constant functions will do the work.

Adding to the statement of the problem that the $F=(f_1,\ldots,f_n)$ function is invertible we get a correct theorem. Maybe other weakening of this condition works.

Problem 4, p. 88.

Last line of problem 4 says "Also show that XY itself is not a vector field." and should say "Also show that XY itself is not always a vector field." There are trivial examples in which XY is a vector field. For example if X is identically zero. There are non-trivial examples too but lets give them after the due day of Homework III because I'm sure you will enjoy finding those examples by your self.