# Is time the real line?

November 22, 2021

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.

Keywords:

*none*