A Lindelöfication

An almost real maximal ideal M of C(X) is a maximal ideal that is either fixed or Z[M] contains a free z-filter which is closed under countable intersection. Using these maximal ideals, we first construct a space λX which is weakly Lindelöf containing X as a dense subspace on which every f∈C(X) with Lindelöf cozero-set can be extended (when this is the case, we say that X is CL-embedded). Next, using this space, we present the largest Lindelöf subspace ΛX of βX in which X is CL-embedded. If X is locally Lindelöf, λX coincides with ΛX and it turns out in this case that ΛX(=λX) is the smallest Lindelöf subspace of βX with compact remainder. Using the structure of ΛX, we also give the smallest realcompact subspace ϒX of βX with compact remainder. Finally the relations between the spaces υX, λX, ϒX, ΛX and βX are investigated and we apply the structures of these spaces to characterize some intersections of free maximal ideals of C(X).