Section Intervals
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 \(\R\text{.}\) 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.
Definition 1.1.
A subset \(I\) of \(\R\) is an interval if for all \(a\text{,}\) \(b\text{,}\) and \(c\) in \(\R\) (allowing for \(a\) or \(b\) to be \(\pm \infty\)) with \(a \lt c \lt b\text{,}\) if \(a\) and \(b\) are in \(I\text{,}\) then \(c\) is in \(I\text{.}\)
With this definition, the set of all real numbers \(x\) satisfying \(0 \lt x \lt 1\) is an interval that we denote by \((0,1)\) (it is important to understand the context — we also use the notation \((0,1)\) to denote an ordered pair). The general notation we use for intervals is the following:
\((a,b) = \{t \in \R \mid a \lt t \lt b\}\) (\(a\) or \(b\) could be \(\pm \infty\))
\([a,b) = \{t \in \R \mid a \leq t \lt b\}\) (\(b\) could be \(\pm \infty\))
\((a,b] = \{t \in \R \mid a \lt t \leq b\}\) (\(a\) could be \(\pm \infty\))
\([a,b] = \{t \in \R \mid a \leq t \leq b\}\text{.}\)
In this notation, \(\R = (-\infty, \infty)\text{.}\) 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 \(a \lt b\) in the interval notation. This implies that \((1,1)\) is an interval. Since there are no real numbers larger than \(1\) and less than \(1\text{,}\) \(\emptyset = (1,1)\) 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 \(\emptyset\) 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.
