This paper is devoted to discussing the topological structure of the arrow of
time. In the literature, it is often accepted that its algebraic and
topological structures are that of a one-dimensional Euclidean space
$\mathbb{E}^1$, although a critical review on the subject is not easy to be
found. Hence, leveraging on an operational approach, we collect evidences to
identify it structurally as a normed vector space $(\mathbb{Q}, |\cdot|)$, and
take a leap of abstraction to complete it, up to isometries, to the real line.
During the development of the paper, the space-time is recognized as a
fibration, with the fibers being the sets of simultaneous events. The
corresponding topology is also exposed: open sets naturally arise within our
construction, showing that the classical space-time is non-Hausdorff. The
transition from relativistic to classical regimes is explored too.
