Quasi-open map

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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[edit]

A function between topological spaces and is quasi-open if, for any non-empty open set , the interior of in is non-empty.[1][2]

Properties[edit]

Let be a function such that X and Y are topological spaces.

  • If is continuous, it need not be quasi-open. Conversely if is quasi-open, it need not be continuous.[1]
  • If is open, then is quasi-open.[1]
  • If is a local homeomorphism, then is quasi-open.[1]
  • If and are both quasi-open (such that all spaces are topological), then the function composition is quasi-open.[1]

References[edit]

  1. ^ a b c d e f 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.