Let $x$ be any space. Define a relation ${}_{pc}$ on $x$ as follows: $x{}_{pc}y$ if there is a path in $x$

from $x$ to $y.$

$($a$)$ Show that ${}_{pc}$ is an equivalence relation. The equivalence classes for ${}_{pc}$ are called the

path components of $x.$

$($b$)$ Describe the set $\frac{x}{{}_{pc}}$ for the following spaces:

i$.x=S^3$

ii$.x$ is the topologist's sine curve

iii. $x=Z[0,1)$