Quasi-open map

In topology a branch of mathematics, a quasi-open map or quasi-interior map is a function which has similar properties to continuous maps. However, continuous maps and quasi-open maps are not related.^[1]

Definition

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

Properties

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

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

References

1. Kim, Jae Woon (1998). "A Note on Quasi-Open Maps" (PDF). Journal of the Korean Mathematical Society. B: The Pure and Applied Mathematics 5 (1): 1–3. Retrieved October 20, 2011.
2. Blokh, A.; Oversteegen, L.; Tymchatyn, E.D. (2006). "On almost one-to-one maps". Trans. Amer. Math. Soc.