One point compactification:

Let $X$ be a locally compact, non-compact Hausdorff space and $\widehat{X} = X\cup\{\infty\}$ be the one point union of $X$ with an additional point $\infty$. The topology ${\cal T}$ consists of all the open subsets in $X$ as well as all the subsets of the form $\{\infty\}\cup(X-K)$, where $K$ ranges over all the compact subsets of $X$. The following theorem summarizes the properties of $\widehat{X}$ and the proof is left for the reader.