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 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. Further information: quasicomponent is union of 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.