In this work, we look at the dynamics of four different spaces, the interval, the unit circle, subshifts of finite type and compact countable sets. We put our emphasis on chaotic dynamical system and exhibit sufficient conditions for the system on the interval, the unit circle and subshifts of finite type to be chaotic in three different types of chaos. On the interval, we reveal two weak conditions’s role as a fast track to chaotic behavior. We also explain how a strong dense periodicity property influences chaotic behavior of dynamics on the interval, the unit circle and subshifts of finite type. Finally we show how dynamics property of compact countable sets effecting the structure of the sets.