We prove the main theorem on one-point compactification rigorously. Namely, we prove: A topological space $X$ is locally compact and Hausdorff if and only if there exists a topological space $Y$ such that
1) $X$ is subspace of $Y$.
2) The complement of $X$ in $Y$ is singletone set and
3) $Y$ is compact and Hausdorff.
Then we define (one-point) compactification.
At the end, I have given two exercises. Please try to solve.
