Differentiable Manifold minus point not compact

Question

Suppose $X$ is an $n$-dimensional for differentiable manifold for $n \geq 1$: in our definition this is a second countable Hausdorff space with a maximal differentiable atlas. If $p \in X$ is a point, we are asked to prove that $X \backslash \{p\}$ is not compact. I don't really see why. I have had a hint to look at a chart around $p$ which somehow can be related to $D^n$ minus a point, where $D^n$ is a compact $n$-dimensional ball. However, it is required to use the Hausdorff property for this, I cannot really see how to use it here though. Any hints?
Thanks in advance!

Answer 1

In a chart around $p$, consider the set of all closed balls centered at $p$ of radius $1/n$ for some integer $n$. Every finite intersection of sets in this collection is nonempty but the intersection of all of them is empty. Since the manifold is Hausdorff and closed balls in the chart are compact, they are closed in the manifold. Once a point is removed, the punctured balls are closed. Thus the intersection of all these sets being empty implies the punctured manifold is not compact.

As suggested by Najib, an equivalent formulation is to consider the collection of complements of the punctured balls in the manifold, which constitutes an open cover, and the open cover contains no finite subcover because all of the sets in a finite subcover would be contained in the largest one, which does not contain every point in the punctured manifold.