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In physics, one of the ways to solve problems is to find a similar mathematical model which describes the problem, and that leads to my question:

By experiments we familiar with the space of time, we know how to find the right time when a rocket reached the height of $10$ kilometers, or what is the exact time when an object will fall to the ground in free fall.

To solve these problems we are using infinitesimal calculus, but the usage of "calculus tools" is allowed when the topology on the space of ($1$ or more dimensional) time is the euclidean topology.

How do we know the time is not a discrete space? After all, we usually say "one moment please", if the space of time is built out of "moments", then the time is discrete, therefore compact sets are useless because now the compact sets are only finite sets, which means that the extreme value theorem is nothing, and the time is not even connected.

Why only by experimants we can determine the topology of time? Or is it just a convention as a mathematical model?

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    $\begingroup$ I'm not sure I understand your question. The only reason we know anything about the physical universe is through experiment. $\endgroup$
    – J. Murray
    Mar 19, 2021 at 22:11
  • $\begingroup$ I mean how do we know that the topology is the euclidean topology even in very very very small (or infinitesimal) intervals of time. $\endgroup$
    – Or Shahar
    Mar 19, 2021 at 22:18

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I mean how do we know that the topology is the euclidean topology even in very very very small (or infinitesimal) intervals of time.

We don't. Spacetime could very well be discrete, or maybe our entire conception of space and time is ultimately wrong and needs to be revised.

The reason we treat spacetime as continuous is because no experiment has ever indicated that spacetime is discrete. That doesn't mean that it isn't, just that if it is, the scale at which the continuum model fails is beyond the reach of our current experiments.

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