If totally disconnectedness does not imply the discrete topology, then what is wrong with my argument? Assume that $X$ is a totally disconnected space. Then every two-point set is disconnected, which implies that every singleton is open in the topology of $X$ (because the one-point subsets of two-point sets form a separation). Isn't the collection of those singletons a basis for the discrete topology? So isn't the topology discrete?
 A: Totally disconnected means that no two points are in the same connected component. So for $x,y\in X$ there is no connected set containing both points. In particular the set $\{x,y\}$ is not connected, thus is discrete, thus each point is open in the subspace topology of this set. That does not mean that it is open in $X$.
A: Note that being totally disconnected means that the only connected nonempty subsets are the singletons.  To see that, for example, $\{ a , b \}$ is not connected, it suffices to find open subsets $U , V \subseteq X$ with the following properties:


*

*$U \cap \{ a,b \} = \{ a \}$;

*$V \cap \{ a,b \} = \{ b \}$;


This would not imply, however, that either $\{ a \}$ are $\{ b \}$ are open subsets of $X$.  This only means that $\{ a \}$ and $\{ b \}$ are open subsets of the subspace $\{ a,b \}$ of $X$.

To give more details with regards to an actual example, consider the rationals $\mathbb{Q}$ as a subspace of the real line.  Given any subset $A \subseteq \mathbb{Q}$ of size $> 1$ pick $p,q \in A$ with $p < q$.  Then there is an irrational number $x$ such that $p < x < q$.  Note that $U = ( - \infty , x ) \cap \mathbb{Q}$ and $V = ( x , + \infty )  \cap \mathbb{Q}$ are open subsets of $\mathbb{Q}$ which have the following properties:


*

*$U \cap A \neq \emptyset \neq V \cap A$;

*$A \subseteq U \cap V$; and

*$( U \cap V ) \cap A = \emptyset$.


This demonstrates that $A$ cannot be a connected subset of $\mathbb{Q}$; and in general, the only nonempty connected subsets of $\mathbb{Q}$ are the singletons.
Note, however all nonempty open subsets of $\mathbb{Q}$ are infinite.
