Metric spaces of semicontinuous functions (Q1893643)

A metric in the spaces of bounded upper-semicontinuous real functions defined on the \(n\)-dimensional interval is introduced -- useful in some applications -- with which the spaces \(MS(I)\) and \(DS(I)\) are separable. Here \(MS(I)\) denotes the set of all real bounded upper-semicontinuous functions on the interval \(I\) and \(DS(I)\) denotes the set of all nonnegative bounded upper-semicontinuous functions on the interval \(I.\) Moreover, the space \(DS(I)\) is complete while \(MS(I)\) is not complete.