equivalent

Let $\mathfrak M$ be a class of mappings. Two spaces X and Y are said to be equivalent with respect to $\mathfrak M$ (shortly $\mathfrak M$- equivalent) if there are two mappings, both in $\mathfrak M$, one from X onto Y and the other from Y onto X. If $\mathfrak M$ means the class of monotone mappings, we say that X and Y are monotonely equivalent.