Preview Activity 15.1.
Let \((X, \tau)\) be a topological space and \(A\) a nonempty subset of \(X\text{.}\) It is reasonable to use the open sets in \(X\) to define open sets in \(A\text{.}\) More specifically, we might consider a subset \(O_A\) of \(A\) to be open in \(A\) if \(O_A\) is the intersection of \(A\) with some open set in \(X\text{,}\) as illustrated in Figure 15.1. With this in mind we define \(\tau_A\) as
\begin{equation*}
\tau_A = \{O \cap A \mid O \in \tau\}\text{.}
\end{equation*}
(a)
Show that \(\tau_A\) is a topology on \(A\text{.}\) The result of item (1) is that any subset of a topological space \((X,\tau)\) is also a topological space with topology \(\tau_A\text{.}\)
Definition 15.2.
Let \((X,\tau)\) be a topological space. A subspace of \((X,\tau)\) is a nonempty subset \(A\) of \(X\) together with the topology
\begin{equation*}
\tau_A = \{O \cap A \mid O \in \tau\}\text{.}
\end{equation*}
(b)
For each of the following, \(X\) is a topological space and \(\tau\) is a topology on \(X\text{.}\)
(i)
Let \(X= \{a,b,c,d\}\) and \(\tau = \{\emptyset, \{a\}, \{b\}, \{a,b\}, X \}\text{.}\) Consider the subset \(A=\{b,c\}\) and list the open sets in the subspace topology \(\tau_A\text{.}\) Now consider \(Z = \{a,b\}\text{.}\) What is the name of the subspace topology \(\tau_Z\) on this subset of \(X\text{?}\)
(ii)
Consider \(X=\R\) with \(\tau\) the indiscrete topology. What are the open sets in the subspace topology on \([1,2]\text{?}\) Now generalize to any nonempty set in the indiscrete topology.
(iii)
Let \(X = \{a,b,c,d,e,f,g,h,i\}\) with \(\tau\) the discrete topology. What are the open sets in the subspace topology on \(\{a,b,d\}\text{.}\) Now generalize to any nonempty set in the discrete topology.
(iv)
Let \(X= \{a,b,c,d,e,f\}\) with \(\tau = \{\emptyset,\{a\}, \{c,d\}, \{a,c,d\}, \{b,c,d,e,f\}, X\}\text{.}\) What are the open sets in the subspace \(A = \{a, b, e\}\text{?}\) Is every open set in \(A\) an open set in \(X\text{?}\) Explain.
(v)
Let \(X=\Z\) with \(\tau = \tau_{FC}\) the finite complement topology. What are the open sets in the subspace topology on \(A = \{0,19, 37, 5284\}\text{?}\) Can you generalize this to the subspace topology on any finite subset of \(\Z\text{?}\)
(vi)
Let \(X=\Z\) with \(\tau = \tau_{FC}\) the finite complement topology. What are the open sets in the subspace topology on the even integers? Can you generalize this to the subspace topology on any infinite subset of \(\Z\text{?}\)
