# 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 FX(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 FX(U) to be the set of all real-valued functions which are constant on open intervals forming U. The family F = {FX(U):U 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 FX(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 FX(U)). Clearly, (X,FX) is not a smooth manifold. Even taking X as any T2 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.