Quasicomponent

Definition

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 clopen subsets containing $x$.

Equivalence of definitions

Further information: equivalence of definitions of quasicomponent

Related notions

• Connected component is a notion that coincides with quasicomponent for a locally connected space (and for many other kinds of spaces). 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.