The map $p: R \rightarrow S^1$ given by the equation $p(x) = (\cos 2\pi x,\sin 2\pi x)$
And we are to consider the subset U of $S^1$ consisting of those points having positive first coordinate. Then they say that the set $p^{-1}(U)$, consisting of those points x for which $\cos 2\pi x$ is positive is the union of the intervals
$$V_n = \left(n-\frac{1}{4}, n + \frac{1}{4}\right)$$
This is what it need to have explained: Further they say that, restricted to the closed interval $\bar{V_n}$, p carries $\bar{V_n}$ surjectively onto $\bar{U}$ and $V_n$ to U, by the intermediate value theorem. How is this?
It further says that since $\bar{V_n}$ is compact, $p \restriction{\bar{V_n}}$ is a homeomorphism of $\bar{V_n}$ with $\bar{U}$. How is this?