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?