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My questions are :

  1. Why do we commonly use certain usual topologies rather than others ? For example the usual topology on the real numbers, the topology of uniform convergence, the compact-open topology...
  2. In which sense are these topologies natural ? Is it because they use the information gained from the situation in the most efficient/general way (for example, making use of a norm in the most general way, or making use of the topology of an intermediate space in the case of function spaces, or making use of an algebraic operation, or an equivalence relation, ...) ?
  3. Since it is easy to construct topologies (by taking the topology generated by a family of subsets), why do we almost always use $\mathbb{R}$ with its usual topology and function spaces with their usual ones.

Any link towards an article or a book would be much appreciated.

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    $\begingroup$ This is rather like asking "why do we use the usual addition on $\mathbb R$ rather than a different one?" If we equip $\mathbb R$ with a different topology, then, as a topological space, it isn't $\mathbb R$ anymore. That is, once we call it "$\mathbb R$ with a different topology", we might as well also call it "$\mathbb C$ with a different topology", or "the first uncountable ordinal with a different topology". $\endgroup$
    – LSpice
    Commented Sep 5, 2016 at 20:50
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    $\begingroup$ In short, because they're useful in other areas of mathematics (e.g. metric geometry, functional analysis, ...). The notions in topology provide a unified way of thinking about all types of convergence, wherever they arise in mathematics. So the "natural" topologies, etc., are ones that arise in other fields of mathematics, e.g. the compact-open topology arising as the topology induced by the sup norm when the spaces in question are suitably restricted. $\endgroup$
    – Neal
    Commented Sep 5, 2016 at 20:52
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    $\begingroup$ @LSpice The first uncountable cardinal might be smaller than the continuum. $\endgroup$
    – Michael Greinecker
    Commented Sep 5, 2016 at 20:52

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Q: Why do we use certain operations rather than others? For example the usual addition for the real numbers?

A: Because, if we used some other addition it would not be what we call "the real numbers". That is: "the real numbers" is not just a naked set. It is that set together with certain structure.

Similarly, with some other topology, it would not be what we call "the real numbers".

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  • $\begingroup$ More specifically, «real numbers» are the completion of $\mathbb Q$ with respect to the usual, normed topology. There are other completions of $\mathbb Q$. So real numbers arise by construction with a topology: although their set could be equipped with another, it wouldn't be natural. $\endgroup$
    – Loïc Teyssier
    Commented Sep 6, 2016 at 7:47

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