Last time we defined open sets in metric spaces. This time we're going to turn it around and define closed sets. Open sets are defined based on neighborhoods of points in the set. Unsurprisingly, closed sets are as well, but the neighborhoods are used differently.
A point, p, is a limit point of a set if every neighborhood of p contains a point other than p in the set. Importantly, p does not have to be a member of the set in order to be a limit point of the set. Consider the set of all real numbers less than 1. 1 is not itself a member of this set. Let's look at neighborhoods of 1. The neighborhood of radius r around 1 includes all real numbers x such that 1−r<x<1+r. Since this clearly includes points less than 1 for any positive value for r, 1 is a limit point of this set. Of course, if the set were the points less than or equal to 1, 1 would still be a limit point of the set, but it would also be a member of the set.
If p is a limit point of a set, every neighborhood of p must contain at least one point other than p in the set. This implies that every neighborhood of p in fact contains an infinite number of points in the set. Assume that the number of points in some neighborhood which are in the set is finite. (The neighborhood of course includes all the points in the space within some distance of p, but we are only interested in the points in the neighborhood which are also in the set.) Then there is a closest point to p in the set, and the distance from p to the closest point is some positive real number. Then the neighborhood of p with a radius smaller than that distance doesn't contain any points in the set other than p. Since by assumption every neighborhood of p must contain points in the set, the neighborhood must contain an infinite number of points from the set.
Moving on, let's start with the set of real numbers less than 1 again, and add to it the single point 2. It's obvious that any neighborhood of 2 with radius less than 1 doesn't include any other points in the set. Therefore, 2 is not a limit point of the set. Since 2 is member of the set but is not a limit point of the set, it is called an isolated point.
Let's look at the set of rational numbers of the form 1/n, where n is any natural number. This set has a limit point, 0. For any positive value of r, it's possible to pick an n>1/r, and the point 1/n is less than r, and so the neighborhood of 0 with radius r includes a point in the set. However, 0 is not itself a member of the set. Furthermore, every point in the set is an isolated point, because given two adjacent points, you can always find a radius less than the distance between them, so every point is in a little island neighborhood containing only itself.
If you were feeling poetic, you could imagine the set as an infinite set of lonely points, each reaching out to the one goal that could connect them all, the limit point 0, but no point ever connecting to any of the points around it. But it's probably better to just think of them as emotionless rational numbers.
Anyway, this gives us a definition of closed sets. A set is closed if every limit point of the set is also a member of the set. Notice the set can include, or entirely consist of, isolated points. There just can't be any limit points of the set which are outside the set itself. A finite set can't have any limit points, and therefore, if the set has any points (it's nonempty), every point in the set is an isolated point, and, the set is closed.