Connected Component

Definition

Consider a Topological Space $(X,\mathcal{T})$. The connected component $C_x$ of a point $x\in X$ is the union of all connected subsets of $X$ that contain $x$. It is the unique largest (with respect to $\subseteq$) connected subset of $X$ that contains $x$.

Facts

Each point $x\in X$ is contained in exactly one component.

Given points $x,y\in X$, their components $C_x,C_y$ are the same or disjoint. $\forall x,y\in X,C_x=C_y\text{or}{C}_x\cap C_y=\varnothing$

Every connected subset in $X$ is contained in some component.

Components of $X$ are closed sets in $X$.

$X$ is a Connected Space if and only if $X$ consists of a single component.

If $C$ is a component of $X$ and $A,B$ are separable sets, then $C$ is a subset of $A$ or $B$.