limit point compact
Limit point compactness says the space is too cramped for an infinite set of points to spread out without crowding somewhere: any infinite collection of points must pile up arbitrarily close to at least one point of the space. There is no room for infinitely many points to keep their distance from one another forever.
Formally, a space X is limit point compact if every infinite subset of X has a limit point in X — a point p every neighborhood of which contains some point of the subset other than p itself. This is a direct abstraction of the Bolzano–Weierstrass theorem, which is why the property is sometimes named after it.
Limit point compactness is the third member of the compactness family, alongside (cover) compactness and sequential compactness. In general topological spaces all three can differ, and limit point compact is the weakest of the lot in the sense that cover-compact spaces are always limit point compact but not conversely. The happy resolution, again, is metric spaces: there compact, sequentially compact, and limit point compact are equivalent, and the distinctions only matter once you leave the metric world behind.
Inside [0, 1] the infinite set {1, 1/2, 1/3, 1/4, ...} has the limit point 0: every neighborhood of 0, say (-epsilon, epsilon), swallows all but finitely many of the points 1/n. So this infinite set crowds at 0, illustrating that [0, 1] is limit point compact; the infinite set {1, 2, 3, ...} in R has no limit point at all, so R is not.
{1/n} crowds at 0 inside [0,1]; the unbounded set {1,2,3,...} in R has no limit point.