New classes of compact-type spaces

Being motivated by the notions of kappa-Frechet--Urysohn spaces and k'-spaces introduced by Arhangelskii, the notion of sequential spaces and the study of Ascoli spaces, we introduce three new classes of compact-type spaces. They are defined by the possibility to attain each or some of boundary points x of an open set U by a sequence in U converging to x or by a relatively compact subset A subset U such that x is in the closure of A. 

Relationships of the introduced classes with the classical classes (as, for example, the classes of kappa-Frechet--Urysohn spaces, (sequentially) Ascoli spaces, k_IR-spaces, s_IR-spaces etc.) are given. We characterize these new classes of spaces and study them with respect to taking products, subspaces and quotients. In particular, we give new characterizations of kappa-Frechet--Urysohn spaces and show that each feathered topological group is kappa-Frechet--Urysohn. We describe locally compact abelian groups which endowed with the Bohr topology belong to one of the aforementioned classes. Numerous examples are presented.