An additivity theorem for uniformly continuous functions (Q1763603)

A metric space \(X\) is said to be straight (2-straight) if for any finite (resp. two set) closed cover of \(X\) any continuous real-valued function on \(X\) is uniformly continuous if and only if its restriction to each of the closed sets of the cover is uniformly continuous. It is shown that 2-straight is equivalent to straight. For a locally connected metric space, \(X\), it is also shown that \(X\) is straight if and only if \(X\) is uniformly locally connected i.e. for all \(\varepsilon> 0\), there is a \(\delta> 0\) such that any two points less than \(\delta\) apart lie in a connected set of diameter less than \(\varepsilon\). Moreover, for a totally disconnected space, \(X\), (i.e. a space for which the intersection of all open sets containing a point is that point) it is proved that \(X\) is straight if and only if \(X\) is a UC space i.e. if and only if all real-valued functions on \(X\) are uniformly continuous.