Quasi-open map

A function ${\displaystyle f:X\to Y}$ between topological spaces ${\displaystyle X}$ and ${\displaystyle Y}$ is quasi-open if, for any non-empty open set ${\displaystyle U\subset X}$, the interior of ${\displaystyle f(U)}$ in ${\displaystyle Y}$ is non-empty.[1][2]

Let ${\displaystyle f:X\to Y}$ be a function such that X and Y are topological spaces.

• If ${\displaystyle f}$ is continuous, it need not be quasi-open. Conversely if ${\displaystyle f}$ is quasi-open, it need not be continuous.[1]
• If ${\displaystyle f}$ is open, then ${\displaystyle f}$ is quasi-open.[1]
• If ${\displaystyle f}$ is a local homeomorphism, then ${\displaystyle f}$ is quasi-open.[1]
• If ${\displaystyle f:X\to Y}$ and ${\displaystyle g:Y\to Z}$ are both quasi-open (such that all spaces are topological), then the function composition ${\displaystyle h=g\circ f:X\to Z}$ is quasi-open.[1]