Algebras of Cauchy continuous maps (Q799968)

For \(F\subset R^ X\) a new Cauchy structure \(\gamma_ F\) on F is defined; it is coarser than the structure of uniform convergence (equal if F consists of bounded functions) and finer than the structure induced by the continuous convergence. It is proved that F is the set of all Cauchy-continuous maps on X with respect to a Cauchy structure on X iff F is a function algebra containing constants which is complete in \(\gamma_ F\). If F is a sublattice of \(R^ X\) containing constants, then the set of bounded functions from F is sequentially dense in \((F,\gamma_ F)\).