Totally disconnected space

In topology and related branches of mathematics, a totally disconnected space is a topological space that is maximally disconnected, in the sense that it has no non-trivial connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) are connected; in a totally disconnected space, these are the "only" connected proper subsets.
An important example of a totally disconnected space is the Cantor set. Another example, playing a key role in algebraic number theory, is the field of "p"-adic numbers.
Definition.
A topological space formula_1 is totally disconnected if the connected components in formula_1 are the one-point sets. Analogously, a topological space formula_1 is totally path-disconnected if all path-components in formula_1 are the one-point sets.
Another closely related notion is that of a totally separated space, i.e. a space where quasicomponents are singletons. Equivalently, a topological space 
formula_1 is totally separated space if and only if for every formula_6, the intersection of all clopen neighborhoods of formula_7 is the singleton formula_8. Equivalently, for each pair of distinct points formula_9, there is a pair of disjoint open neighborhoods formula_10 of formula_11 such that formula_12.
Every totally separated space is evidently totally disconnected but the converse is false even for metric spaces. For instance, take formula_1 to be the Cantor's teepee, which is the Knaster–Kuratowski fan with the apex removed. Then formula_1 is totally disconnected but its quasicomponents are not singletons. For locally compact Hausdorff spaces the two notions (totally disconnected and totally separated) are equivalent.
Unfortunately in the literature (for instance ), totally disconnected spaces are sometimes called hereditarily disconnected while the terminology totally disconnected is used for totally separated spaces. 
Examples.
The following are examples of totally disconnected spaces:
Constructing a totally disconnected space.
Let formula_1 be an arbitrary topological space. Let formula_17 if and only if formula_18 (where formula_19 denotes the largest connected subset containing formula_7). This is obviously an equivalence relation whose equivalence classes are the connected components of formula_1. Endow formula_22 with the quotient topology, i.e. the finest topology making the map formula_23 continuous. With a little bit of effort we can see that formula_22 is totally disconnected. We also have the following universal property: if formula_25 a continuous map to a totally disconnected space formula_26, then there exists a "unique" continuous map formula_27 with formula_28.