Quasi-topological spaces were proposed by Edwin Spanier as a substitute for Top which has properties convenient for the purposes of algebraic topology. In particular, they form a complete and cocomplete cartesian closed category.
Quasi-topological spaces today seem to be regarded mostly as a historical curiosity, perhaps because working topologists were never comfortable with the set-theoretic issues that accompany them. In retrospect, however, they are an impressive testament to the conceptual insight of Spanier into ideas of topos theory which were at the time (early 1960’s) barely in the air, and even not quite born yet (being an early example of quasitopos, whose name perhaps derives from Spanier’s notion).
A quasi-topological space is a (small-set valued) concrete sheaf on .
The (super-large) category of quasi-topological spaces is a quasitopos (although this is not immediately obvious for size reasons — in particular, it is probably not a Grothendieck quasitopos). In particular, it is a locally cartesian closed category.