In the last fifteen years a large
number of classes of isocompact spaces were investigated by many mathematicians.
In this paper we introduce a new large class (i.e. the class of k-neat spaces) of
isocompact spaces. This class contains all of the following classes: neighborhood
ℱ-spaces, spaces satisfying property 𝜃L, weakly [ω1,∞)r-refinable spaces,
δ𝜃-penetrable spaces and pure spaces. Other properties of this class are also
investigated. For example we show that an ω1-compact ω1-neat T1-space is
a-realcompact and k-neatness is an inverse invariant of maps under some conditions.
In the last section we consider compactness of isocompact spaces having a countably
compact dense subset.
