Pseudo-open mapping

A continuous mapping $ f : X \rightarrow Y $ such that for every point $ y \in Y $ and any neighbourhood $ U $ of the set $ f ^ { - 1 } y $ in $ X $ it is always true that $ y \in \mathop{\rm Int} f U $( here $ \mathop{\rm Int} f U $ is the set of all interior points of $ f U $ with respect to $ Y $).

Comments

It is also called a hereditarily quotient mapping, because a mapping $ f: X \rightarrow Y $ is pseudo-open if and only if for every $ B \subseteq Y $ the corestriction $ f _ {B} : f ^ { - 1 } [ B] \rightarrow B $ is a quotient mapping.

References

[a1]	R. Engelking, "General topology" , Heldermann (1989)