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.