Recall　that　a　space　X　is　a　c-semistratifiable　(CSS)　space,　if　the　compact　sets　of　X　are　G　δ-sets　in　a　uniform　way.　In　this　note,　we　introduce　another　class　of　spaces,　denoting　it　by　k-c-semistratifiable　(k-CSS),　which　generalizes　the　concept　of　c-semistratifiable.　We　discuss　some　properties　of　k-c-semistratifiable　spaces.　We　prove　that　a　T　2-space　X　is　a　k-c-semistratifiable　space　if　and　only　if　X　has　a　g　function　which　satisfies　the　following　conditions:　(1)　For　each　x　2　X,　{x}　=　∩{g(x,　n):　n　2　ℕ}　and　g(x,　n　+　1)　{square　image　of　or　equal　to}　g(x,　n)　for　each　n　∈　ℕ.　(2)　If　a　sequence　{xn}　n∈ℕ　of　X　converges　to　a　point　x　∈　X　and　y　n　∈　g(x　n,　n)　for　each　n　∈　ℕ,　then　for　any　convergent　subsequence　{y　nk}　k∈ℕ　of　{y　n}　n∈ℕ　we　have　that　{y　nk}　k∈ℕ　converges　to　x.　By　the　above　characterization,　we　show　that　if　X　is　a　submesocompact　locally　k-csemistratifiable　space,　then　X　is　a　k-c-semistratifible　space,　and　the　countable　product　of　k-c-semistratifiable　spaces　is　a　k-c-semistratifiable　space.　If　X　=　∪{Int(X　n):　n　∈　ℕ}　and　X　n　is　a　closed　k-c-semistratifiable　space　for　each　n,　then　X　is　a　k-c-semistratifiable　space.　In　the　last　part　of　this　note,　we　show　that　if　X　=　∪{X　n:　n　∈　ℕ}　and　X　n　is　a　closed　strong　β-space　for　each　n　∈　ℕ,　then　X　is　a　strong　β-space.