Topology: An Inquiry-Based Approach

The real numbers have a special property that allows us to, among other things, define the distance between a point and a set in a metric space. It also allows us to define distances between subsets of certain types of metric spaces, which creates a whole new metric space whose elements are the subsets of the metric space. We will examine that property of the real numbers in this activity.

This might lead us to suggest that the distance from a point \(x\) to a set \(A\text{,}\) denoted by \(d(x,A)\) is the minimum distance from the point to any point in the set, or \(d(x,A) = \min\{d_E(x,a) \mid a \in A\}\text{.}\)

What if we changed the set \(A\) to be the open interval \((-1,0)\text{?}\) What then should \(d(x,A)\) be, or should this distance even exist? If we think of the distance between a point and a set as measuring how far we have to travel from the point until we reach the set, then in the case of \(x=1\) and \(A=(-1,0)\text{,}\) as soon as we travel a distance more than 1 from \(x\) in the direction of \(A\text{,}\) we reach the set \(A\text{.}\) So we might intuitively say that \(d_E(x,(-1,0)) = 1\) as well. But we cannot define this distance as a distance from \(x\) to a point in \(A\) since \(0 \notin A\text{.}\) We need a different way to formulate the notion of a distance from a point to a set.

In a case like this, with \(x=1\) and \(A = (-1,0)\text{,}\) we can examine the set \(T=\{d_E(x,a) \mid a \in A\}\) and notice some facts about this set. For example, the set \(T\) is a subset of the nonnegative real numbers. Also, in this example there are no numbers in \(T\) that are smaller than 1. Because of this property, we will call the number 1 a lower bound for \(T\text{.}\) More generally,

If a subset \(S\) of \(\R\) has a lower bound, we say that \(S\) is bounded below. So the set \(T = \{d_E(1,a) \mid a \in (-1,0)\}\) is bounded below by 1. The set \(T\) is also bounded below by 0.5 and 0. In fact, any number less than 1 is a lower bound for \(T\text{.}\) The critical idea, though, is that no number larger than 1 is a lower bound for \(T\text{.}\) Because of this we call 1 a greatest lower bound of \(T\text{.}\) More generally,

A greatest lower bound is also called an infimum. We might now use this idea of a greatest lower bound to define the distance between \(1\) and \(A = (-1,0)\) as the greatest lower bound of the set \(\{d_E(1,a) \mid a \in (-1,0)\}\text{.}\) However, there are questions we need to address before we can do so. One question is whether or not every nonempty subset of \(\R\) that is bounded below has an infimum. The answer to this question is yes, and we will take this result as an axiom of the real number system (often called the completeness axiom).