Intervals and Neighbourhoods

The real line. Every real number corresponds to a unique point on a line (and vice versa); we call this line the real line. Values increase to the right, decrease to the left, and since there is no gap anywhere on the line, between any two reals there are infinitely many more reals.

Definition of interval. A subset $I\subseteq R$ is an interval if (i) $I$ has at least two elements, and (ii) whenever $a,b\in I$ and $a<c<b$, then $c\in I$ too (no "holes"). Geometrically, intervals are exactly the rays and line segments of the real line. Sets like $N$, $W$, the odd integers, the even integers, or the primes are not intervals -- they have gaps (e.g. no integer lies strictly between $3$ and $4$, but genuine intervals never skip real numbers that way).

Finite vs. infinite intervals. A finite interval has two finite endpoints (though it still contains infinitely many real numbers between them); an infinite interval extends to $-\infty$ and/or $\infty$ (which are not numbers, just symbols marking "no bound"). A finite interval is closed if it contains both endpoints, open if it contains neither.

The eight standard interval types (for $a,b\in R,\ a<b$; "$\circ$" marks an excluded endpoint, "$\bullet$" an included one):

and $(-\infty,\infty)=R$, the whole real line. …