4.13 Path-Connectedness
Definition 4.13.1 (Path). Let $X$ be a topological space, $x, y \in X$, then a path from $x$ to $y$ is a mapping $f \in C([0, 1]; X)$ such that $f(0) = x$ and $f(1) = y$.
Definition 4.13.2 (Path-Connected). Let $X$ be a topological space, then $X$ is path-connected if for every $x, y \in X$, there exists a path from $x$ to $y$.
Proposition 4.13.3. Let $X$ be a path-connected space, then $X$ is connected.
Proof. Let $U, V \subset [0, 1]$ be open with $[0, 1] = U \cup V$. If $\sup U = \sup V$, then $\sup U = \sup V = 1$ and $U, V \in \cn^{o}(1)$, so $U \cap V \ne \emptyset$. If $\sup U < \sup V \le 1$, then $x \not\in U$ and $x \in V$. In which case, $V \in \cn^{o}(x)$ and $V \cap U \ne \emptyset$. Therefore $[0, 1]$ is connected.
Fix $x \in X$. For any $y \in X$, let $f_{y} \in C([0, 1]; X)$ be a path from $x$ to $y$, then $f_{y}([0, 1])$ is connected with $x, y \in f_{y}([0, 1])$ by Proposition 4.12.3. By Proposition 4.12.2,
is connected.$\square$
Proposition 4.13.4. Let $X$ be a topological space, $\seqi{A}$ be path-connected sets with $\bigcap_{i \in I}A_{i} \ne \emptyset$, then $\bigcup_{i \in I}A_{i}$ is connected.
Proof. Let $x \in \bigcap_{i \in I}A_{i}$, $i, j \in I$, $y \in A_{i}$, and $z \in A_{j}$. By connectedness of $A_{i}$ and $A_{j}$, there exists paths $f \in C([0, 1]; X)$ from $y$ to $x$, and $g \in C([0, 1]; X)$ from $x$ to $z$. Thus the concatenation
is a path from $y$ to $z$.$\square$
Definition 4.13.5 (Path Component). Let $X$ be a topological space and $A \subset X$ be path-connected, then there exists a unique path-connected set $C \supset A$ such that for any $C' \supset A$ path-connected, $C \supset C'$. The set $C$ is the path-component of $A$.
Proof. Let $C$ be the union of all path-connected sets containing $A$, then $C$ is path-connected by Proposition 4.13.4 and the maximum path-connected set containing $A$ by definition.$\square$