Connected components

Definition

1. Equivalence relation

   Let $X$ be a topological space. We define a relation $\sim$ on $X$ by declaring $x\sim y$ if there is $Y\subseteq X$ connected, such that $x,y\in Y$.

2. Connected components

   One can prove that $\sim$ is an equivalence relation. The equivalence classes are called connected components of $X$.

Properties

1. Teorema

   The connected components of $X$ have the following properties:

   • They are connected, disjoint subsets of $X$ with union $X$.
   • Each connected nonempty subset of $X$ intersects exactly one of them.
   • They are closed, and if there are finitely many components, they are also open.

Path components

Definition

1. A new equivalence relation

   We define a relation $\sim$ on the topological space $X$, by setting $x\sim y$ if there is a path from $x$ to $y$.

2. Path components

   One can prove that $\sim$ is an equivalence relation. The equivalance classes are called path components of $X$.

Properties

1. Teorema

   The path components of $X$ have the following properties:

   • They are path-connected, disjoint subsets of $X$ with union $X$.
   • Each nonempty path-connected subset of $X$ intersects exactly one of them.

The topologist’s sine curve

Local conectedness

Definition

1. Locally connected space
   • $X$ is locally connected at $x\in X$ if for any neighborhood $U$ of $x$, there is a connected neighborhood of $x$ contained in $U$.
   • If $X$ is locally connected at every point, then we say that $X$ is locally connected.
2. Locally path connected space
   • $X$ is locally path connected at $x\in X$ if for any neighborhood $U$ of $x$, there is a path connected neighborhood of $x$ contained in $U$.
   • If $X$ is locally path connected at every point, then we say that $X$ is locally path connected.

Examples

1. :B_exampleblock:
   • Any interval in $\mathbb{R}$ is connected and locally connected.
   • The subspace $[0,1]\cup [2,3]\subseteq \mathbb{R}$ is locally connected and not connected.
   • The topologist’s sine curve is connected but not locally connected.
   • $\mathbb{Q}$ as a subspace of $\mathbb{R}$ is neither connected nor locally connected.

2. Teorema

   A space $X$ is locally connected if and only if for every open set $U$ of $X$, each component of $U$ is open in $X$.

3. Proof
   • Suppose that $X$ is locally connected, and let $C$ be a component of the open set $U\subseteq X$.
   • Let $x\in C$, and choose $V$ a connected neighborhood of $x$ with $V\subseteq U$.
   • Since $V$ is connected, it must be contained in $C$.
   • Now suppose the components of open sets in $X$ are open.
   • Let $x\in X$ and a neighborhood $U$ of $x$. If $C$ is the component of $U$ that contains $x$, then $C$ is a connected neighborhood of $x$.\qed

The proof of the following theorem is similar and left as an exercise.

1. Teorema

   A space $X$ is locally path connected if and only if for every open set $U$ of $X$, each path component of $U$ is open in $X$.

Components and path components

1. Teorema
   • For any space $X$, each path component of $X$ is contained in a component of $X$.
   • If $X$ is locally path connected, the components and the path components are the same.
2. Proof
   • Let $C$ be a component of $X$, let $x\in C$, and let $P$ be the path component of $X$ containg $x$. Since $P$ is connected, $P\subseteq C$.
   • Suppose now that $X$ is locally path connected, and we wish to prove $P=C$. Assume that $P\subsetneq C$.
3. Proof (continuation)
   • Let $S$ be the union of all the path components of $X$ different from $P$ that intersect $C$. Since such components must lie in $C$, we have $C=P\cup S$.
   • Since $X$ is locally path connected, each path component of $X$ is open in $X$. Hence $P$ and $S$ are open, disjoint and nonempty sets with union $C$. This contradicts that $C$ is connected.\qed