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 → Y between topological spaces X and Y is quasi-open if, for any non-empty open set U ⊆ X, the interior of f ('U) in Y is non-empty.^[1][2]

Properties

Let ${\displaystyle f:X\to Y}$ be a map between 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]
• The composition of two quasi-open maps is again quasi-open.^{[note 1][1]}

See also

• Almost open map – Map that satisfies a condition similar to that of being an open map.
• Closed graph – Graph of a map closed in the product spacePages displaying short descriptions of redirect targets
• Closed linear operator
• Open and closed maps – A function that sends open (resp. closed) subsets to open (resp. closed) subsets
• Proper map – Map between topological spaces with the property that the preimage of every compact is compact
• Quotient map (topology) – Topological space constructionPages displaying short descriptions of redirect targets

Notes

1. This means that 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 ${\displaystyle g\circ f:X\to Z}$ is quasi-open.

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. Archived from the original (PDF) on March 4, 2016. Retrieved October 20, 2011.
2. Blokh, A.; Oversteegen, L.; Tymchatyn, E.D. (2006). "On almost one-to-one maps". Trans. Amer. Math. Soc. 358 (11): 5003–5015. doi:10.1090/s0002-9947-06-03922-5.