Certain sequences with compact closure

This paper deals with a question which is stated by quite simple definitions. A sequence { x n } in a space X is called a β-sequence if every subsequence of it has a cluster point in X. The closure of the sequence { x n } means the closure of { x n : n ∈ ω } in X. Here we consider the question when a β-sequence has compact closure. We give several answers to this question.