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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=\mathbb{B}} be endowed with the ordinary topology (thus, it is Hausdorff and second countable). Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} be an arbitrary connected open set in $X$ (that is, an interval). Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_X(U)} consists of all functions identically equal to constant. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_X(U)} to be the set of all real-valued functions which are constant on open intervals forming Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} . The family Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F=\{F_X(U):U\text{ 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g=h \circ f} (if $g$ is given, then we define $h(x)=g$ for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , if $f$ is given, then we take arbitrary smooth Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.