LEC ANA2 14

Definition 337.1(Path connectedness).

Let $X$ be a metric space. Let $x,y\in X$. A path from $x$ to $y$ is a continuous function $\gamma :[0,1]\to X$ such that $\gamma (0)=x$, $\gamma (1)=y$. If there exists a path between $x$ and $y$, the points are said to be path connected. $S\subseteq X$ is called path connected if every pair of points $x,y\in S$ are path connected.

Path connectedness is a equivalence relation.

Proposition 337.2(Proposition).

Path connected $⟹$ connected.

[!Examples]
These are path connected:

1. ${\mathbb{R}}^n\setminus \{\text{countable set}\}$
2. open balls in NLS

[!Proposition]
Let $X$ be a NLS. If $U\subseteq X$ is open and connected, it is path connected.

[!Proof]-
Let $x_0\in U$. Let $E=\{x\in U:x,x_0\text{ are path connected}\}$. It suffices to prove $E$ is both open and closed.

Let $x\in E$. Since $U$ is open, there exists an open ball centered at $x$ contained in $U$; since open balls in an NLS are path connected, the open ball also lies in $E$, hence $E$ is connected. It can be similarly shown that $E^c$ is open.

Proposition 337.3(Proposition).

Let $E$ be connected. Then, any $F$ such that $E\subseteq F\subseteq \overline{E}$ is path connected.

Example 337.4(Example).

Standard example of a connected but not path connected set.