Some topological properties of $C[0,1]$

Question

Consider the space $X=C[0,1]$, with its usual 'sup-norm' topology. Let $$S=\Big\{f \in X: \int_0^1 f(t)\; dt=0 \Big\}$$

Then which of the following are true?

a) $S$ is closed

b) $S$ is connected

c) $S$ is compact

I know it is closed, since it is the inverse image of $\{0\}$ under $G: f(x) \mapsto \int_0^1 f(x)\;dx$

What about the rests? Can I have a hint?

Answer 1

Hint for $(b)$, its even path connected. Hint for $(c)$, try to find a sequence of functions in $S$ that have no subsequence that is convergent in the sup-norm.

Answer 2

$S$ is a linear subspace of $X$. Using this property, you can show that $S$ is connected (since it is convex) and that $S$ fails to be compact (since it is unbounded).