Restricted Metric Subspaces as Topological Subspaces

# Restricted Metric Subspaces as Topological Subspaces

Recall from the Subspace Topologies page that if $(X, \tau)$ is a topological space and $A \subseteq X$ then the subspace topology on $A$ is defined to be:

(1)
\begin{align} \quad \tau_A = \{ A \cap U : U \in \tau \} \end{align}

Furthermore, if $A$ is a subset of the metric space $(X, \tau)$ then we will often say that the term "topological subspace $A$" refers to the set $A$ with the subspace topology $\tau_A$, i.e., $(A, \tau_A)$.

We will now look at an important class of topological subspaces that arises from metric spaces and metric subspaces.

Consider a metric space $(M, d)$ where $d$ is the metric defined on $M$. Then $M$ is also a topological space where the topology $\tau_M$ on $M$ is defined to be the unions of open balls with respect to the metric $d$.

If we take any nonempty subset $S \subseteq M$ then $(S, d)$ also forms a metric space with the metric $d$ being restricted to $S$. What will the topology $\tau_S$ on $S$ be then? Conveniently enough, we will see that:

(2)
\begin{align} \quad \tau_S = \{ S \cap U : U \in \tau_M \} \end{align}

In other words, for every open set $V \in \tau_S$ there exists a $U \in \tau_M$ such that:

(3)
\begin{align} \quad V = S \cap U \end{align}

This fact is often proven when looking at metric subspaces and we will prove an analogous theorem regarding topological spaces and topological subspaces soon.