Quasi-open map Generalization of open map in topology
In topology , a branch of mathematics, a quasi-open map (also called quasi-interior map ) is a function that generalizes the notion of open map .
Definition
A function f : X → Y topological spaces is called quasi-open if, for any nonempty open set U ⊆ X interior of f ( U ) Y quasi-interior map .
Properties
Let f : X → Y topological spaces .
If f f : ℝ → ℝ f ( x ) = 0
Conversely, if f f : ℝ → ℝ f ( x ) = x x < 0 f ( x ) = x + 1 x ≥ 0
If f open , then f f : ℝ → ℝ , x ↦ sin ( x )
If f local homeomorphism , then f
The composition of two quasi-open maps is quasi-open. See also
Almost open map – Map that satisfies a condition similar to that of being an open map Closed graph – Property of functions in topology Pages displaying short descriptions of redirect targets Closed linear operator – Linear operator whose graph is closed Open and closed maps – Functions that send open (resp. closed) subsets to open (resp. closed) subsets Proper map – Mathematical map between topological spaces Quotient map (topology) – Topological space construction Pages displaying short descriptions of redirect targets Notes
↑ This means that if f : X → Y g : Y → Z g ∘ f : X → Z
References
↑ Mardešić, Sibe; Papić, Pavle (1962). "Continuous images of ordered compacta, the Suslin property and dyadic compacta" (PDF) . Period. Math.-Phys. Astron., II. Ser . 17 : 3– 22. Zbl 0119.17906 . 1 2 3 Kao, Kuo Shih (1983). "A note on M1-spaces". Pacific Journal of Mathematics . 108 (1): 121– 128. doi :10.2140/pjm.1983.108.121 . Zbl 0487.54029 . ↑ 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 ↑ 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 .