Section The Subspace Topology
In our preview activity, we saw that the intersection of the open sets in a topological space \(X\) with any nonempty subset \(A\) of \(X\) forms a topology for \(A\text{.}\) We then have \(A\) as a subspace of \(X\text{.}\)
The topology
\(\tau_A\) in
Definition 15.2 is called the
subspace topology, the
induced topology, or the
relative topology. In our preview activity we saw that sets that are open in a subspace
\(A\) of a topological space
\(X\) need not be open in
\(X\text{.}\) So we call the sets in
\(\tau_A\) relatively open.
Once we have defined relatively open sets, we can then consider how to define relatively closed sets.
Activity 15.2.
Let \((X, \tau)\) be a topological space, and let \(A\) be a subset of \(X\text{.}\)
(a)
Recall that a subset of a topological space is closed if its complement is open. Given that \((A, \tau_A)\) is a topological space, how is a closed set in \(A\) defined? Such a set will be called relatively closed.
(b)
Recall that a subset \(U\) of \(A\) is relatively open if and only if \(U = A \cap O\) for some open subset of \(X\text{.}\) With this in mind, how might we expect a relatively closed set in \(A\) to be related to a closed set in \(X\text{?}\) State and prove a theorem for this result.
