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Quasi-open map

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Function that maps non-empty open sets to sets that have non-empty interior in its codomain

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.

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.

Properties

Let f : X → Y {\displaystyle f:X\to Y} be a map between topological spaces.

  • If f {\displaystyle f} is continuous, it need not be quasi-open. Conversely if f {\displaystyle f} is quasi-open, it need not be continuous.
  • If f {\displaystyle f} is open, then f {\displaystyle f} is quasi-open.
  • If f {\displaystyle f} is a local homeomorphism, then f {\displaystyle f} is quasi-open.
  • The composition of two quasi-open maps is again quasi-open.

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 f : X → Y {\displaystyle f:X\to Y} and g : Y → Z {\displaystyle g:Y\to Z} are both quasi-open (such that all spaces are topological), then the function composition g ∘ f : X → Z {\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.


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