We want language for recording the elements of some set that appear at regular intervals of time.
Let $A$ be a set. An observation sequence (or time series) in $A$ is a mapping $a: \Z \to A$.
If $A = \R ^d$ in which case we call $a$ a vector-valued observation sequence or vector-valued time series.
Examples include temperature readings, seismograph readings, sunspot numbers, export quantities, and so on.
The most basic assumption we can make is that the observation sequence $a$ exhibits some regularity over time. This assumption is naturally related to the assumption of time-invariance in physical laws.1