This is a result of point-set topology. There is a one-point compactification for any locally compact Hausdorff space. I put the definitions you need (for a terse but rigorous run at it) down below as a starter for some words in case you care to go into details. Don't worry if it scares you! If you study mathematics, this will all become clear in time. I just felt like writing. This is all in the following notes; you should go there for a fully fledged proof of the result (p. 60, 3.7.1): http://folk.uio.no/rognes/kurs/mat4500h10/topology.pdf
The statement of the theorem is as follows:
One-point compactification: Let $X$ be a locally compact Hausdorff space, and $Y=X\cup\{\infty\}$. Give $Y$ the topology consisting of the open sets of $X$ together with, for each compact subset $K$ of $X$, sets of the form $Y-K$. Then $Y$ is a compact Hausdorff space. Proof is in the notes above.
Topological space: A topology on a set $X$ is a collection $T$ of subsets of $X$ called open sets such that 1) $T$ contains $X$ and the empty set $\emptyset$, 2) $T$ is closed under arbitrary unions, and 3) $T$ is closed under finite intersections. We say that a subset of $X$ is closed if its compliment is an open set (note that $X$ and $\emptyset$ are both open and closed subsets of $X$). We call the pair $(X,T)$ a topological space, and often denote it by just $X$ and think of the topology $T$ as given.
Here closed under finite intersections means if you take a finite collection of open sets, their intersection is an open set. Similarily for closure under arbitrary unions. This is very different from what we called a closed subset of $X$.
Neighbourhood: If $x\in X$ we say that a neighbourhood of $x$ is any subset of $X$ containing an open set containing $x$. (Beware that some authors define a neighbourhood of $x$ simply as an open set containing $x$; we call these open neighbourhoods.)
Hausdorff: We say that a space $X$ is Hausdorff if for each pair of distinct points $x,y$, there exist disjoint open neighbourhoods of $x$ and $y$.
Compactness: An open covering of $X$ is a collection of open subsets of $X$ such that their union is all of $X$ (note that with these definitions $T$ is the biggest possible open covering of $X$). A space $X$ is called compact if each open covering has a finite open subcovering.
Subspaces: We will want to talk about compact subsets $K$ of $X$, and we can by viewing them as subspaces of $X$ in the subspace topology. That is the set $K$ together with the topology induced by $X$; open sets of $K$ are sets of the form $U\cap K$ with $U$ open in $X$.
Compactness, as you might have guessed, is a very nice property. It is supposed to capture our intuition of being closed, small and bounded. In fact, any compact subset of a Hausdorff space is closed in the technical sense. Also, any compact subset of a metric space has a well-defined diameter. It also gives a finiteness condition that can help us out in all sorts of ways, and it has some nice formal properties: the product of an arbitrary family of compact spaces (with the product topology) is compact (Tychonoff); the image under a continuous map of a compact space, is compact.
A closed and bounded interval of the real line is compact, and (to some, more importantly) all spheres are compact!
Local compactness: We say that a space $X$ is locally compact at $x\in X$ if there is a compact neighbourhood of $x$ (that is, there is a compact subset $K\subseteq X$ and an open subset $V\subseteq K$, with $x\in V\subseteq K\subseteq X$).