Let $M$ be a compact manifold of positive dimension and let $p \in M$. Show that $M$ is homeomorphic to the one-point compactification of $M \setminus \{p\}$
I will let $\Gamma = (M\setminus\{p\}, \mathcal{T}^*)$ denote the one-point compactification of $M \setminus \{p\}$.
There are a few things I can say off the bat about $M$ and $\Gamma$. Note that since $M$ is compact for any open set $U$ in $M$ we have $M \setminus U$ to be compact.
Observe that $M \setminus \{p\}$ is open in $M$ and hence the subspace topology on $M \setminus \{p\}$ contains all open sets in $M$ of the form $V \cap M \setminus\{p\}$ for some open $V$ in $M$.
Since $M \setminus\{p\}$ is Hausdorff every compact subset of $M \setminus \{p\}$ is closed in $M \setminus \{p\}$. Taking the complement of the open sets of the form $V \cap M \setminus \{p\}$ for some open $V$ in $M$, which is $(M\setminus \{p\}) \setminus (V \cap M \setminus \{p\}) =M \setminus \{p\} \setminus V \ \cup \ ((M \setminus \{p\}) \setminus (M \setminus \{p\})) = \left(M \setminus \{p\}\right) \setminus V $ gives us all the compact sets of $M \setminus \{p\}$.
Hence $\mathcal{T}^* = \{ V \cap M \setminus \{p\}\ \ | \ V \text{ is open in } M\} \cup \{ \left(V \cup \{\infty\}\right) \ | \ V \text{ is open in } M\}$
Now that we have a workable description of the topology on $\Gamma$ we should be able to construct a homeomorphism $f : M \to \Gamma$
So I tried to define $f$ by the obvious bijection $$ f(x)= \begin{cases} x \ \ \ \ \text{if} \ \ x \in M \setminus \{p\}\\ \infty \ \ \text{if} \ \ x =p\\ \end{cases} $$
But the problem is then that for any $V$ open in $M$ we have $f^{-1}[V \cup \{\infty\}] = V \cup \{p\}$ which isn't necessarily open. So I think my candidate for the homeomorphism is incorrect.
Any hints on how I should construct this homeomorphism?
There is a theorem (as stated in the comments by Henno Brandsma) that makes this a two-liner; it states that for a locally compact Hausdorff space $X$ and a compact Hausdorff space $Y$, there holds $Y\cong X^+$ whenever there exists $b\in Y$ such that $Y\setminus\{b\}\cong X$.
Since $M$ is a manifold, it is locally compact and since it is Hausdorff, its open subspaces are locally compact, therefore $M\setminus\{p\}$ is locally compact. The choice of point $b$ above is obvious and the seeked homeomorphism is the identity map.