The fundamental group of a half disk ( a neighborhood of a boundary point) can't be detected

Question

In the lecture A professor said something I don't quite understand. We need to show that no interior point of a surfacce is a boundary point. We were showing that by proving the neighborhood $U$ of a interior point $x$ is not homotopy equivalent to the neighborhood $V$ of a boundary point y. Now here is the thing I don't understand: $\pi1(U)$ is certainly trivial and with $x$ removed is $\mathbb{Z}$. But the professor said the foundamental group of $V$ could not be detected and with $y$ removed is still a half disk. Could someone help me here? Perhaps I heard it wrong?