Dispersive space

hereditarily disconnected space

A topological space which contains no connected sets with more than one point.

Comments

Some authors call spaces as above totally disconnected. However, a space is commonly called totally disconnected if for all $x\notin y$ in $X$ there is a closed and open set $C$ such that $x\in C$ and $y\notin C$. A closed and open set is also called a clopen set.

See also Totally-disconnected space.

References

[a1]	K. Kuratowski, "Topology" , 1–2 , Acad. Press (1966–1968) (Translated from French)
[a2]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)