A topological space (X,T) is said to be quasi compact if every open covering has a finite refinement. Let f be a fuction from (X,T) to another topological space (Y,T') then f is said to be quasi compact if inverse image of every quasi compact open subset of Y is a quasi compact subset of X.
Recently I saw another definition of quasi compactness in the research paper On Door Spaces by Julian Dontchev which says that a function f from a topological space (X,T) to another topological space (Y,T') is called quasi compact if it satisfies the following condition:
Let U be an open subset of X such that inverse image of f(U) is U itself then f(U) is open in Y. My doubt is whether this two definitions are equivalent.If yes kindly post the proof .