The Time Scales Calculus

A time scale (which is a special case of a measure chain, see Chapter 8) is an arbitrary nonempty closed subset of the real numbers. Thus $$ \mathbb{R}, \mathbb{Z}, \mathbb{N}, \mathbb{N}_0 , $$ i.e., the real numbers, the integers, the natural numbers, and the nonnegative integers are examples of time scales, as are $$ [0,1] \cup [2,3], [0,1] \cup \mathbb{N}, and the Cantor set, $$ while $$ \mathbb{Q}, \mathbb{R}\backslash \mathbb{Q}, \mathbb{C}, (0,1), $$ i.e., the rational numbers, the irrational numbers, the complex numbers, and the open interval between 0 and 1, are not time scales. Throughout this book we will denote a time scale by the symbol $$ \mathbb{T} $$ . We assume throughout that a time scale $$ \mathbb{T} $$ has the topology that it inherits from the real numbers with the standard topology.