a. This set is clearly disconnected because Determinant map is a continuous function from $\mathcal{O}_n(\mathbb{R})$ to the discrete set $\{1, -1\}$
c. This set can be thought as $\mathbb{R^2}-\mathbb{Q^2}$ which is connected as poved here Proving $\mathbb R^2 \setminus \mathbb Q^2$ is connected
b. I'm confused with this option and can't think of any possible ways to justify. Please give some hint. Thank you.
Hints for (b):
Let $f,g \in S$.
Let $p:[0,1]\to C[0,1]$ be defined by $$p(t)=(1-t)f+tg$$
Show that $p$ is continuous.
Show that the image of $p$ is a subset of $S$.
Note that $p(0)=f$, and $p(1)=g$.
Deduce that $S$ is path-connected.