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For any non-empty set $X$, we can think of multiple topologies on $X$. That is, topologies are not uniquely determined by $X$. For example, the metric induced by $p$-norm and discrete metric yield different topologies on the same set $X$.

But when we talk about the topology on the set of real numbers (especially in analysis), it seems that the metric used for the construction of the topology is tacitly fixed to $p$-norm. For example, the topology on real numbers is induced by the absolute function (as a $p$-norm for scalars).

My questions are 1) whether I correctly understand the meaning of topologies, and 2) when we deal with real numbers, do we use only one topology that is induced by $p$-norm?

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  • $\begingroup$ Well, there's this: en.wikipedia.org/wiki/Lower_limit_topology $\endgroup$
    – jwimberley
    Jan 7, 2021 at 14:20
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    $\begingroup$ This is a broad question that is beyond topology. You said do we use only one topology that is induced by $p$-norm? I will answer by asking you a question. Who is we? If you think to applied mathematics, then yes the usual topology (the one induced by the absolute value) is the on mostly frequently used. But for topologists studying other topologies is of interest. In particular as it enables to uncover specific properties like Haussdorf, normal... For sure the only word is too strong! $\endgroup$ Jan 7, 2021 at 14:30
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    $\begingroup$ Although one might not appreciate it from the very start, the topology of $\mathbb{R}$ is more natural than its metrics or norms. Every linearly ordered set has a "natural" topology, generated by the open intervals. The fact that all the $p$-norms induce this topology is thus not surprising. Also all finite dimensional normed spaces have the same topology. $\endgroup$ Jan 7, 2021 at 14:35
  • $\begingroup$ For a definition of Topologies, see my answer to this :math.stackexchange.com/questions/3977196/… $\endgroup$ Jan 10, 2021 at 9:27

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  1. Yes, you do understand correctly that there can be several topologies on the same set. That being said, a lot of them are rarely used, as they are either boring or too special. This leads to the second point.

  2. There are loads of other topologies on $\mathbb{R}$. For example, in algebraic geometry people will mostly use the Zariski-topology. Also the discrete topology exists on $\mathbb{R}$, just as on any other set. But in differential geometry for example you will almost always encounter the "standard-topology" on $\mathbb{R}$. Same is true for most of applied mathematics. So to answer your second question: No, we do not use only the "standard-topology" on $\mathbb{R}$. It depends largely on who "we" is.

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