Theorem 2.4:

(i) The collection of sets ${\cal T}$ is a topology on $\widehat{X}$.

(ii) The family of sets $\widehat{X} - K$, where $K$ ranges over all compact subsets of $X$, forms a neighborhood base of $\infty$.

(iii) $X$ with the given topology is an open dense subset of $\widehat{X}$.

(iv) The space $\widehat{X}$ is compact.