one-point-compactification is a topological space
Sophia Terry 
Let $X$ be a local compact Hausdorff space endowed with a topology $\tau_X$ and consider $X_{\infty}=X\coprod\{\infty\}$ as a set. Let $$\tau_{X_\infty}:=\tau_X\cup \{U\subseteq X_\infty:\infty\in U\;\text{ and}\; X\setminus U\;\text{ is a compact subset of}\; X\}.$$ Claim: $\tau_{X_\infty}$ is a topology on $X_{\infty}$.
I want to prove: If $U_i\in \tau_{X_\infty}$ for $i\in I$, then $\bigcup_{i\in I} U_i\in\tau_{X_\infty} $ If $\infty\notin \bigcup_{i\in I} U_i$, it is $\bigcup_{i\in I} U_i\in \tau_X\subseteq \tau_{X_\infty}$. Now let $\infty\in \bigcup_{i\in I} U_i$ and write $\bigcup_{i\in I} U_i=( \bigcup_{i\in I,\infty\notin U_i} U_i)\cup ( \bigcup_{i\in I,\infty\in U_i} U_i)$. From the first case it is $\bigcup_{i\in I,\infty\in U_i} U_i\in \tau_{X_\infty}$, but I don't know how to continue.
Now let $U_1,U_2\in \tau_{X_\infty}$, I want to prove: $U_1\cap U_2\in \tau_{X_\infty}$. If $\infty\notin U_1$ for i=1 and i=2, it's clear. If $\infty\in U_1$, but $\infty\notin U_2$, $X\setminus U_2$ is compact in $X$ and $U_1\in \tau_{X_\infty}$. I don't know how to continue.
This is only where I'm stuck. I appreciate your help.
$\endgroup$1 Answer
$\begingroup$HINT: For the union, let $J=\{i\in I:\infty\in U_i\}$; then
$$X\setminus\bigcup_{j\in J}U_j=\bigcap_{j\in J}(X\setminus U_j)\;,$$
which is an intersection of compact subsets of $X$.
For the intersection, suppose that $\infty\in U_1\setminus U_2$. Then $\infty\notin U_1\cap U_2$, so you need to prove that $U_1\cap U_2\in\tau_X$. Let $V=U_1\setminus\{\infty\}=U_1\cap X$; clearly $U_1\cap U_2=V\cap U_2$, so you’ll be done if you can show that $V\in\tau_X$. To do this, note that $X\setminus V=X_\infty\setminus U_1$, which is compact. Now use the fact that $X$ is Hausdorff.
$\endgroup$ 15
