Doubt on the need for Topological Manifolds I happen to be trying to motivate physically and intuitively the need to use topological spaces.
So without using emerging concepts such as differentiable manifolds or extremely formal concepts as continuity I ask you:

What is so fundamental about physics that motivates and requires us to use topological spaces?

 A: 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.
A: There have been a number of proposals motivated by quantum gravity to use non-Hausdorff manifolds to describe timeline splitting (as in the many worlds interpretation) and avoid time travel paradoxes.
https://www.researchgate.net/publication/334223872_Interpreting_non-Hausdorff_generalized_manifolds_in_General_Relativity
https://arxiv.org/abs/gr-qc/0505150
A: This is a little more philosophy/mathematics than physics, but there is an alternative approach pioneered by Tim Maudlin that tries to replace point-set topology with a theory about straight lines. He takes the topological manifold approach to lack an intuitive basis, and so he proposes an alternative mathematical theory. It is called the theory of linear-structures. If you are interested, here is the link to the book
https://www.amazon.com/New-Foundations-Physical-Geometry-Structures/dp/0198701306/ref=sr_1_1?dchild=1&keywords=tim+maudlin+linear+structures&qid=1599746275&sr=8-1
A: The simple answer is, that that is what spacetime physically is, a topological space.
It all started when Euler first counted the edges of a polyhedron and came up with his famous formula V - E + F = 2. When polyhedra or graphs drawn on surfaces like toroids, Klein bottles and Moebius strips came under study, they did not match his formula. The study of such surfaces became known as analysis situs and the surfaces as manifolds. Mathematicians then began wondering about solids, spaces and their equivalents in higher dimensions, all now understood as manifolds in n dimensions and the field was renamed topology. Spacetime is just such a 4-manifold.
We don't know its shape. Does time have a beginning or an end or does it all curl round on itself as Stephen Hawking suggested? Might space have a boundary or does it too curl round on itself like the surface of a sphere? If they curl, in what way do they curl? The possible manifolds are just the solutions to the equations of General Relativity, but we don't know which is the right solution.
Having developed this approach, the mathematics began cropping up in all sorts of different places. There is little intuition for this, only the observation that the maths is still described by the same terms, even though it is applied in the context of subnuclear particle scattering interactions or whatever.
