# 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 ${\displaystyle f:X\to Y}$ 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]

## Properties

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]

## 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.