Saying a function is increasing/decreasing does not restrict the definition of the function to just open sets. This property can hold true not only on open sets. For example, the function $ f(x) = x $ defined is increasing, either if it is defined on an open set $ (0,1) $ or a closed one $ [0,1] $, or even $ [0,1) $.

The choice of open sets in the definition of increasing/decreasing may appear due to the nature of the text you're reading it from. If the text discusses differentiation, defining functions on open sets allows us to ignore treating the derivate of the function at the boundary of the set where it would be ill defined (in the traditional sense of the derivative), thus making life a little easier.