In topology, a discrete topology \(\tau\) on a set \(X\) is a topology which contains all the subsets of \(X\). One may wonder what is the rational for naming such a topology a discrete topology. The motivation for such a naming can be understood as follows.
First lets understand, what we mean by a discrete set. In the context of a metric-space, by a discrete set, what we mean is given any element in that set we can "separate" it from all other elements in the set by say possibly a small ball (set) around that point. For instance, the set of integers is a discrete set of \(\mathbb{R}\) under our usual metric space structure on \(\mathbb{R}\), since given any integer, \(n\), we can consider balls centered at \(n\) of radius say \(\frac12\). Any two balls are disjoint from one another. Hence, the set of integers is a discrete subset of real numbers. What we have essentially done is for any point in the set of integers we have found a set containing only that particular integer and no other integer. This is the motivation to define a discrete topology. (Note that the set of rationals do not form a discrete subset of the real numbers nor do the reals under our usual metric space structure on \(\mathbb{R}\).)
Now given any set \(X\) if we want it to be a discrete set then given any element \(x \in X\) we need to find a set which contains the element \(x\) "separately" from the rest of the elements. The only possible choice of this set is \(\{x\}\). Hence, if want \(X\) to be a discrete set, then we need to have all sets of the form \(\{x\}\) where \(x \in X\) to it into a discrete set. Now we look at the topology generated by all singleton subsets of a set \(X\) and it makes sense to call this discrete topology. The topology generated by all singleton subsets of a set \(X\) is nothing but the set of all subsets of \(X\) i.e. \(\tau\{\{x\}:x \in X \} = 2^X\). Hence this is called the discrete topology.
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