  We will give a complete description of $\mathcal{I}$, the set of invertible
quasicontinuous functions on the unit circle. After doing this, we will then
classify the path-connected components of $\mathcal{I}$ and show that
$\mathcal{I}$ has uncountably many path-connected components. We will then use
the above classifications to characterize $\mathcal{F}$, the set of Fredholm
operators of the C$^*$-algebra generated by the Toeplitz operators $T_\phi$
with quasicontinuous symbols $\phi$. Then we will classify the path-connected
components of $\mathcal{F}$ and show that $\mathcal{F}$ also has uncountably
many path-connected components.
