Topology: An Inquiry-Based Approach

We will begin with one of the most basic type of set we will encounter — intervals. The open intervals will be important as they will form a basis for the standard topology on $\mathbb{R}\text{.}$ We are likely familiar with intervals from algebra and calculus, sets like $(0,1)\text{,}$ $[2,3)\text{.}$ To really understand intervals, we will need a rigorous definition.

In this notation, $\mathbb{R}=(-\mathrm{\infty },\mathrm{\infty })\text{.}$ Intervals of the form $(a,b)$ are called open intervals, intervals of the form $[a,b]$ are called closed intervals, and intervals of the form $[a,b)$ or $(a,b]$ are half-open (or half-closed) intervals. The reason for this terminology should become more clear as we introduce open and closed sets later.

Note that nothing in the definition indicates that we must have $a<b$ in the interval notation. This implies that $(1,1)$ is an interval. Since there are no real numbers larger than $1$ and less than $1\text{,}$ $\mathrm{\varnothing }=(1,1)$ is an interval. We could also have an interval of the form $[a,a]$ where $a$ is any real number. This means that $\{a\}=[a,a]\text{,}$ and any single point set is an interval. The intervals $\mathrm{\varnothing }$ and $[a,a]$ for any real number $a$ are called degenerate intervals.

One last note about intervals. Some require that $a$ be less than $b$ in the definition of an interval, with the result that there are no degenerate intervals. This is a matter of debate that we won’t get into. In almost all of our work, we will consider only non-degenerate intervals so this won’t be an issue for us.