0708-1300/Errata to Bredon's Book

Errata to Bredn's Book

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

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

Adding to the statement of the problem that the [math]\displaystyle{ F=(f_1,\ldots,f_n) }[/math] function is invertible we get a correct theorem. Maybe other weakening of this condition works.