20 - Path-Connectedness

Path Connectedness

Let $X$ be a metric space then

$X$ is path-connected if
For $a,b\in X$ there is a continuous map $\gamma :[0,1]\to X$ with

$$\gamma (0)=a\text{ and }\gamma (1)=b$$

Path

Let $X$ be a metric space then

Continuous Map $\gamma :[0,1]\to X$ is called a path

Concatenation of Paths

Let $X$ be a metric space with two paths ${\gamma }_1,{\gamma }_2$ such that

$${\gamma }_1(1)={\gamma }_2(0)$$

then concatenation $y_1\star y_2$ of the two paths is defined by

$${\gamma }_1\star {\gamma }_2(t)=\{\begin{array}{ll}{\gamma }_1(2t)& \text{if }0\le t\le \frac{1}{2}\\ & \\ {\gamma }_2(2t-1)& \text{if }\frac{1}{2}\le t\le 1\end{array}$$

Opposite Path

Let $X$ be a metric space with path $\gamma$ then

Opposite path ${\gamma }^{-1}$ is defined by

$${\gamma }^{-1}(t)=\gamma (1-t)$$

Equivalence Relation between Elements of a Metric Space

Let $X$ be a metric space

Define relation $\sim$ on $X$ as follows

$$a\sim b⟺\text{exists path }\gamma :[0,1]\to X\text{ with }\gamma (0)=a\text{ and }\gamma (1)=b$$

then

$$\sim \text{ is an equivalence relation}$$

Proof

$a\sim a$ by $\gamma (t)=a$ which takes constant value $a$
$a\sim b$ implies $b\sim a$ by taking path $\gamma$ from $a$ to $b$ and considering opposite path $g^{-1}$
$a\sim b\text{ and }b\sim c$ implies $a\sim c$ use concatenation of the two paths

Path-Components

Let $X$ be a metric space then

Equivalence classes in which ~ partitions $X$ is called path-components of $X$