Zeitmenge

Zeitmenge, literally “time set” in German, is a mathematical and modeling concept that denotes the domain of time over which a process is defined. It is typically a subset of the real numbers with the usual order, representing instants or intervals of time. In many applications, the Zeitmenge is an interval such as [0, ∞) or [t0, t1], but it may also be a discrete set like the integers, or a union of disjoint intervals, depending on whether time is modeled continuously or discretely.

Functions describing temporal evolution have domain equal to the Zeitmenge: f: T → X, where T is the Zeitmenge and X is the state space. For example, the trajectory t ↦ x(t) maps each allowed time to a state. The choice of T encodes the temporal horizon and regularity of the model; T can be closed, open, half-open, or a union of intervals, and it may be entire ℝ for steady-time models.

Properties: If T ⊆ ℝ is given a topology induced by the real line, notions such as continuity of f depend on T. In continuous-time models, T is often an interval with the standard topology; in discrete-time models, T ≅ ℤ with the discrete topology. The time set also determines measure-theoretic considerations, such as integrating f over time.

Applications: Zeitmengen appear in physics and engineering for modeling time evolution, in control theory and signal processing, in computer science for real-time systems, and in dynamical systems theory as the temporal domain of trajectories. They serve as a formal abstraction for when events occur and how long processes last.

Notes: The term Zeitmenge is primarily used in German-language texts and is often translated as “time set” or “temporal domain” in English. In practice, the concept overlaps with time intervals and time domains used in analysis.