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In [[topology]] a branch of mathematics, a '''quasi-open map''' or '''quasi-interior map''' is a [[function (mathematics)|function]] which has similar properties to [[continuous map]]s. However, continuous maps and quasi-open maps are not related.<ref name="Kim" />

== Definition ==
A function <math>f : X \to Y</math> between [[topological space]]s <math>X</math> and <math>Y</math> is quasi-open if, for any non-empty [[open set]] <math>U \subset X</math>, the [[interior (topology)|interior]] of <math>f(U)</math> in <math>Y</math> is non-empty.<ref name="Kim">{{cite journal|title=A Note on Quasi-Open Maps|first=Jae Woon|last=Kim|journal=Journal of the Korean Mathematical Society|series=B: The Pure and Applied Mathematics|volume=5|number=1|pages=1–3|year=1998|url=http://icms.kaist.ac.kr/mathnet/kms_tex/50115.pdf|format=pdf|accessdate=October 20, 2011}}</ref><ref>{{cite journal|first1=A.|last1=Blokh|first2=L.|last2=Oversteegen|first3=E.D.|last3=Tymchatyn|title=On almost one-to-one maps|journal=Trans. Amer. Math. Soc|year=2006}}</ref>

== Properties ==
Let <math>f: X \to Y</math> be a function such that ''X'' and ''Y'' are topological spaces.
* If <math>f</math> is continuous, it need not be quasi-open. Conversely if <math>f</math> is quasi-open, it need not be continuous.<ref name="Kim" />
* If <math>f</math> is [[open function|open]], then <math>f</math> is quasi-open.<ref name="Kim" />
* If <math>f</math> is a [[local homeomorphism]], then <math>f</math> is quasi-open.<ref name="Kim" />
* If <math>f: X \to Y</math> and <math>g: Y \to Z</math> are both quasi-open (such that all spaces are topological), then the function composition <math>h = g \circ f: X \to Z</math> is quasi-open.<ref name="Kim" />

== References ==
{{Reflist}}

{{DEFAULTSORT:Quasi-Interior}}
[[Category:Topology]]

{{topology-stub}}

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en>Andy Dingley: restore cats