The concept of compactness modulo of ideal was first defined by Newcomb 1967 [11],
also investigated by Rancin 1972 [12] , an extensively studied by Hamlett and Jankovic in
1990 [10] and [2] . Furthermore , Hamlett et al. [12], have also studied the concept under
the term accountably -compact . Abd El-Monsef et al . [8] used semi-open sets to define
the class of -compact spaces and countably -compact spaces . Arafa A. Nasef [1]
introduced -compact space and countably -compact space by using -open sets .
Rodyna A . Hosny [9] studied some types of compactness modulo an ideal called -
compact spaces and countably -compact spaces . Dontchev [4] introduced the notions of
contra-continuity , a function to be contra-continuous if the inverse image of
every open set of is closed in . Recall that a door space [5] is a topological space in
which every subset is either open or closed . Attention : There are some mistakes appear in