A quasi-metric topology compatible with inclusion monotonicity on interval space (Q1371174)

The authors deal with a quasi-metric topology for the interval space. It is well known that the metric topology on the interval space is not compatible with the interval inclusion monotonicity property in the sense that there may exist monotonic functions which are not continuous and vice versa. The authors introduce a quasi-metric topology for the interval space consistent with the real line topology and whose continuous functions are exactly the monotonic ones. The mentioned quasi-metric \(d\) is characterized by the following axioms: (1) \(d(x,x)=0\); (2) \(d(x,z)\leq d(x,y)+d(y,z)\); (3) \(d(x,y)=d(y,x)=0\) implies \(x=y\).