Doubt on the need for Topological Manifolds

Measurement instruments are not infinitely precise, however It is possibile to distinguish objects using them. This is possible when the precision of them permits it. The precision of an instrument around measured values is the physical corresponding of a neighborhood of a point. The fact that two measures can be distinguished by means of sufficiently precise instruments (though not infinitely precise) corresponds to the mathematical fact (Hausdorff property) that two distinct points, e.g., on the real line, admit corresponding neighbourhoods with empty intersection. I stress that I am referring to physical quantities that can be described in a continuous way, in the sense that improving the precision of the instruments I always find new distinct values and no point structure pops out (evidently all this may eventually reveal to be just an approximation valid to some extent only: the last word on our mathematical models to describe what exists is of physics). In this way, standard notions of topology, more precisely Hausdorff topology, arise naturally and quite generally in the physical context. When the set of physical entities you want to describe are also determined by using mutually compatible coordinate systems, the notion of manifold (topological or differentiable depending on your requirements on the mutual compatibility of the used coordinates) naturally enters the play. The notion of topological manifold actually also assumes another technical requirement on the topology (a countable topological basis giving rise to the so called paracompactness property). This hypothesis is difficult to physically justify in elementary contexts and in fact, also in mathematics, it is not always required.

A fundamental context concerns the physical representation of events. They are represented in terms of spatial position and time location. To measure that information you use real instruments, rulers and clocks. The above discussion naturally leads to a basic representation in terms of a Hausdorff space, before introducing other more sophisticated structures, where the neighbourhoods are defined by the precision of instruments.

However there are many other cases, think of the representation of the equilibrium states of a thermodynamical system.

I stress that I mentioned only elementary and intuitive contexts where topology naturally enters. Considering more advanced subjects of physics, different types of topologies arise. Hausdorff property and second countability cease to be relevant in some contexts essentially of quantum nature. QFT, Quantum Gravity, but also QM. There are classifications of classes of entangled states, in finite dimensional Hilbert spaces, which refer to the Zariski topology. As is known, that topology is not Hausdorff.