# Dispersive space

2010 Mathematics Subject Classification: Primary: 54D05 [MSN][ZBL]

hereditarily disconnected space

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

Some authors call spaces as above totally disconnected. However, a space is commonly called totally disconnected if for all $x\neq y$ 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.