Quasicomponent

Definition in terms of equivalence relation
Consider the following relation $$\! \sim$$ on a topological space $$X$$. For points $$a,b \in X$$, we say $$a \sim b$$ if it is not possible to write $$X$$ as a union of disjoint open subsets $$U,V$$ with $$a \in U, b \in V$$.

This relation is an equivalence relation and the equivalence classes in $$X$$ under the relation are termed the quasicomponents of $$X$$.

Definition as intersection of clopen subsets
For a topological space $$X$$, the quasicomponent of a point $$x \in X$$ is defined as the intersection of all the defining ingredient::clopen subsets containing $$x$$.

Related notions

 * Connected component is a maximal non-empty connected subset. In general, each quasicomponent is a union of connected components. In other words, the equivalence relation defining quasicomponents is coarser than the equivalence relation defining connected components.
 * Any open connected component is a quasicomponent. In particular, a locally connected space is a space in which all connected components are open, and hence the connected components are all open and hence coincide with the quasicomponents.