quasicomponent

Let $X$ be a topological space. Define a relation $\sim$ on $X$ as follows: $x\sim y$ if there is no partition of $X$ into disjoint open sets $U$ and $V$ such that $U\cup V=X$, $x\in U$ and $y\in V$.

This is an equivalence relation on $X$. The equivalence classes are called the quasicomponents of $X$.