Section The Distance from a Point to a Set
Metrics are used to establish separation between objects. Topological spaces can be placed into different categories based on how well certain types of sets can be separated. We have defined metrics as measuring distances between points in a metric space, and in this activity we extend that idea to measure the distance between a point and a subset in a metric space. However, there are two questions we need to address before we can do so. The first we mentioned in our preview activity. We will assume the completeness axiom of the reals, that is that any subset of \(\R\) that is bounded below always has a greatest lower bound. The second question is whether or not a greatest lower bound is unique.
Activity 5.2.
Let \(S\) be a subset of \(\R\) that is bounded below, and assume that \(S\) has a greatest lower bound. In this activity we will show that the infimum of \(S\) is unique.
(a)
What method can we use to prove that there is only one greatest lower bound for \(S\text{?}\)
(b)
Suppose \(m\) and \(m'\) are both greatest lower bounds for \(S\text{.}\) Why are \(m\) and \(m'\) both lower bounds for \(S\text{?}\)
(c)
What two things does the second property of a greatest lower bound tell us about the relationship between \(m\) and \(m'\text{?}\)
(d)
Why must the greatest lower bound of \(S\) be unique?
With the existence and uniqueness of greatest lower bounds considered, we can now say that any nonempty subset \(S\) of \(\R\) that is bounded below has a unique greatest lower bound. We use the notation \(\glb(S)\) (or \(\inf(S)\) for infimum of \(S\)) for the greatest lower bound of \(S\text{.}\) There is also a least upper bound (\(\lub(S)\text{,}\) or \(\sup(S)\) for supremum ) of a subset \(S\) of \(\R\) that is bounded above.
Now we can formally define the distance between a point and a subset in a metric space.
Definition 5.3.
Let \((X,d)\) be a metric space, let \(x \in X\text{,}\) and let \(A\) be a nonempty subset of \(X\text{.}\) The distance from \(x\) to \(A\) is
\begin{equation*}
\inf\{d(x,a) \mid a \in A\}\text{.}
\end{equation*}
We denote the distance from \(x\) to \(A\) by \(d(x,A)\text{.}\) When calculating these distances, it must be understood what the underlying metric is.
Activity 5.3.
There are a couple of facts about the distance between a point and a set that we examine in this activity. Let \((X,d)\) be a metric space, let \(x \in X\text{,}\) and let \(A\) be a nonempty subset of \(X\)
(a)
Why must \(d(x,A)\) exist?
(b)
If \(d(x,A) = 0\text{,}\) must \(x \in A\text{?}\)
