Definition
Totally disconnected space
A topological space whose connected subsets have at most one point.
A topological space is totally disconnected if every connected subset has at most one point.
Equivalent formulation
Every connected component is a singleton. The empty space is also totally disconnected.
Examples
Discrete spaces, the rationals with their usual subspace topology, and the Cantor set are totally disconnected. Total disconnectedness does not imply discreteness.
Locally compact groups
Total disconnectedness is one of the defining conditions of a locally profinite group. The existence of compact open subgroups in that setting is an additional theorem, not part of the definition of total disconnectedness.