Since we define the discrete topology of a set $X$ as the set of all possible subsets of $X$, we are forced to consider individual elements of $X$ as open sets.
My question then is, do we consider individual elements as members of the discrete topology or do we have to consider the sets which only contain a single element as members of the discrete topology?
For example, in the discrete topology of $\mathbb{R}^2$, is the point $(1,1)$ a member of the topology, or would we have to say that the set $\{(1,1)\}$ is a member of the topology?
Not sure if this matters, but the way I understand it, is that for subsets we require more than one element, otherwise it is just a member of the set if that makes sense.
Best Answer
A singleton is also a subset. For any $x \in X, \{x\}$ is a subset of $X$ and hence belongs to the discrete topology.
If $\tau$ is the discrete topology on $X,$ then $\{x\}\in \tau$ but $x \notin \tau.$