A subset of a space is almost countably compact in if for every countable cover of by open subsets of , there exists a finite subfamily of such that . In this paper we investigate the relationship between almost countably compact spaces and relatively almost countably compact subsets, and also study various properties of relatively almost countably compact subsets.
A subspace of a space is almost Lindelöf (strongly almost Lindelöf) in if for every open cover of (of by open subsets of ), there exists a countable subset of such that . In this paper we investigate the relationships between relatively almost Lindelöf subset and relatively strongly almost Lindelöf subset by giving some examples, and also study various properties of relatively almost Lindelöf subsets and relatively strongly almost Lindelöf subsets.
We show, in particular, that every remote point of is a nonnormality point of if is a locally compact Lindelöf separable space without isolated points and .
It is proved that every uncountable -bounded group and every homogeneous space containing a convergent sequence are resolvable. We find some conditions for a topological group topology to be irresolvable and maximal.
We shall prove that under CH every regular meta-Lindelöf -space which is locally has the -property. In addition, we shall prove that a regular submeta-Lindelöf -space is if it is locally and has locally extent at most . Moreover, these results can be extended from the class of locally -spaces to the wider class of -scattered spaces. Also, we shall give a direct proof (without using topological games) of the result shown by Peng [On spaces which are D, linearly D and transitively D, Topology...