Photons as Observers and the Extended Real Number Line Topology I am not a physicist. This is the first question I write in such a forum so if there are remarks on how I wrote it, I'll be happy to edit.
I am originally a mathematician with some interest in physics. I studied the basics of Einstein's formulas in regard to relativity, how space and time changes for an observer looking at an object travelling in a certain speed. The math there breaks  down when the speed is c (speed of light).
But I dared, out on being the non physicist that I am, to ask myself what happens if we regard a photon as an observer and ditch calculus for the topology of extended real number line. There we can divide by 0 and have space and time as singular points.
Simply put:

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*The photon can "measure" all points in space at any given moment.

*The photon can measure anything on the timeline.

Example: As I look at the 2-slits experiment and how we "shoot" one particle at a time, this shouldn't matter to a photon because we can't "fool" it by time differences. The photon does not need to interfere with itself but rather it interferes with another photon the will go through the measurement in the future. As soon as we place our own measurement tool to observe, it does not travel in the speed of light and therefore we get a collapsed result because our "observer" is different, as supported by Einstein's relativity.
Obviously, I expect to be wrong. My question is simple:
Why can't we use another topology other than calculus for Einstein's Relativity?
Why isn't physics allowing this?
 A: Photons as some kind of observer (as if they were like the standard timelike observers in relativity)? As suggested by the discussion linked by @BillyIstiak, there are problems.

*

*For example, what does it mean for a photon "to measure"?
 A clear definition needs to be made or proposed.

For timelike inertial observers, one has (for example) the "radar method" to assign coordinates to all events by sending a radar signal and noting that observer's clock-readings at emission and reception.

Concerning the use of other topological structures in relativity, I'll list some examples below as possible starting points for you. I can't really comment much on them right now... but I merely point out alternatives that have been attempted.

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*Projective geometry (e.g. https://en.wikipedia.org/wiki/Cayley%E2%80%93Klein_metric) and Conformal geometry (e.g. Ehlers-Pirani-Schild EPS, Woodhouse https://doi.org/10.1063/1.1666344 )

*https://en.wikipedia.org/wiki/Spacetime_topology (e.g. Alexandrov, Zeeman, path topology of Hawking-King-McCarthy, Gobel https://doi.org/10.1063/1.522984 , Fullwood https://aip.scitation.org/doi/10.1063/1.529644 , Malament  https://aip.scitation.org/doi/10.1063/1.523436  )

(I hope this list of wikipedia links and DOIs is okay. I could transcribe fuller references, if requested.)
A: Photons as observers
is an imprecise interpretation of the role of the photon. Photons can only be a means of observation. One can observe their emission and absorption. One can observe the transition of the momentum of the photon to subatomic particles. One can measure the collective behaviour of photons in an electromagnetic wave.
The photon has no ontological properties. And thus no room for interpretation that it reaches any entities because of its speed of light.
