Quasi-open map
From HandWiki
Short description: 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.[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 [math]\displaystyle{ f:X\to Y }[/math] be a map between topological spaces.
- If [math]\displaystyle{ f }[/math] is continuous, it need not be quasi-open. Conversely if [math]\displaystyle{ f }[/math] is quasi-open, it need not be continuous.[1]
- If [math]\displaystyle{ f }[/math] is open, then [math]\displaystyle{ f }[/math] is quasi-open.[1]
- If [math]\displaystyle{ f }[/math] is a local homeomorphism, then [math]\displaystyle{ f }[/math] 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 space
- 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
Notes
- ↑ This means that if [math]\displaystyle{ f:X\to Y }[/math] and [math]\displaystyle{ g:Y\to Z }[/math] are both quasi-open (such that all spaces are topological), then the function composition [math]\displaystyle{ g\circ f: X\to Z }[/math] is quasi-open.
References
- ↑ 1.0 1.1 1.2 1.3 1.4 1.5 Kim, Jae Woon (1998). "A Note on Quasi-Open Maps". Journal of the Korean Mathematical Society. B: The Pure and Applied Mathematics 5 (1): 1–3. http://icms.kaist.ac.kr/mathnet/kms_tex/50115.pdf. Retrieved October 20, 2011.
- ↑ 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.
 |  |
