1.4 Metric Subspaces
So far we have defined metric spaces and discussed some of the associated basic concepts. In this
section, given a metric space \((X,d)\), we will discuss properties of the metric spaces that arise from the
metric d and subsets of X. Before we give the next definition, note that if \((X,d)\) is a metric space and \(Y\) is a
subset of \(X\), then all the required axioms for a metric are satisfied when d is restricted to \(Y\times Y\), and so \((Y,d)\) is a
metric space.
Definition 1.4.1. Let \((X,d)\) be a metric space and \(Y\) a subset of \(X\). Then the metric space \((Y,d)\) obtained
by restricting d to \(Y\times Y\) is called the metric subspace of \((X,d)\) induced by \(Y\).
Example 1.4.2. Let \(Y=(0,1]\) be a subset of \(\mathbb {R}\). The metric \((Y,d)\), where \(d\) is the usual metric on \(\mathbb {R}\), is the metric
subspace of \(\mathbb {R}\) induced by \(Y\).
Proposition 1.4.3. Suppose that \((Y,d)\) is a metric subspace of a metric space \((X,d)\) and \(A\) is a subset of \(X\).
If \(A\) is open (closed) in \(X\), and \(A\subseteq Y\), then \(A\) is open (closed) in \(Y\).
Proof. Recall that the open sets of the subspace \(Y\) are exactly the sets \(U\cap Y\) with \(U\) open in \(X\) — that is the definition of the subspace topology.
Suppose \(A\) is open in \(X\) and \(A\subseteq Y\). Then \[A = A\cap Y,\] which exhibits \(A\) as the intersection of a set open in \(X\) with \(Y\), so \(A\) is open in \(Y\).
For the closed case, suppose \(A\) is closed in \(X\) and \(A\subseteq Y\). Then \(X\setminus A\) is open in \(X\), so \[\left (X\setminus A\right )\cap Y = Y\setminus A\] is open in \(Y\), and therefore \(A\) is closed in \(Y\). □
Remark. The converse fails, and the failure is the point of the subspace topology. The set \([0,1)\) is open in the subspace \([0,2]\) of \(\mathbb {R}\), because it equals \((-1,1)\cap [0,2]\), but it is not open in \(\mathbb {R}\). Openness is not a property a set has on its own; it is a property it has relative to a space, and a set can gain or lose it by changing the space it is regarded as sitting in.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.