  In this paper, we study the topological properties and the gap sequences of
Bedford-McMullen sets. First, we introduce a topological condition, the
component separation condition (CSC), and a geometric condition, the
exponential rate condition (ERC). Then we prove that the CSC implies the ERC,
and that both of them are sufficient conditions for obtaining the asymptotic
estimate of gap sequences. We also explore topological properties of
Bedford-McMullen sets and prove that all normal Bedford-McMullen sets with
infinitely many connected components satisfy the CSC, from which we obtain the
asymptotic estimate of the gap sequences of Bedford-McMullen sets without any
restrictions. Finally, we apply our result to Lipschitz equivalence.
