An irreducible surjection is usually defined as a continuous closed surjective map $f:X\rightarrow Y$ such that if for some closed set $C\subset X$ one has $f(C)=Y$ then $C=X$. In my dissertation I used a weaker form of this definition, removing the requirement of these functions being closed. I then defined them as quasi-irreducible surjections. Are there any applications of quasi-irreducible surjection elsewhere?
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In my opinion, the most standard definition of irreducible does not require it to be closed. Some authors do assume closed (such as the book of Porter and Woods, Absolutes and Extensions). In compact spaces, there is no difference of course. In non-compact completely regular spaces, one often deals with perfect irreducible surjections, where a perfect map is closed with compact point-inverses. Bourbaki uses the term "proper map" for perfect map. Some of the papers of V. Fedorchuk from the early 1970's study perfect irreducible maps.
