Summary: In this paper, we look at the chain recurrence set based on the family of open covers of compact Hausdorff spaces. In [Isolated invariant sets and the Morse index. Regional Conference Series in Mathematics. No. 38. Providence, R.I.: American Mathematical Society (AMS). (1978; Zbl 0397.34056); Ergodic Theory Dyn. Syst. 8, 11–26 (1988; Zbl 0687.58033)], C. Conley studied the chain recurrence set in compact metric spaces. We study chain recurrence set with the family of open covers in compact Hausdorff spaces. Even though there are several similarities between the chain recurrent set on compact metric spaces and the chain recurrent set on compact Hausdorff spaces, there is sure the difference on the process of proving because we use open covers.