Section Continuity and Closed Sets
Recall that we showed that a function \(f\) from a metric space \((X,d_X)\) to a metric space \((Y,d_Y)\) is continuous if and only if \(f^{-1}(O)\) is open for every open set \(O\) in \(Y\text{.}\) We might conjecture that a similar result holds for closed sets. Since closed sets are complements of open sets, to make this connection we will want to know how \(X \setminus f^{-1}(B)\) is related to \(f^{-1}(Y \setminus B)\) for \(B \subset Y\text{.}\)
Activity 10.4.
Let \(f\) be a function \(f\) from a metric space \((X,d_X)\) to a metric space \((Y,d_Y)\text{,}\) and let \(B\) be a subset of \(Y\text{.}\)
(a)
Let \(x \in X \setminus f^{-1}(B)\text{.}\)
(i)
What does this tell us about \(f(x)\text{?}\)
(ii)
What can we conclude about the relationship between \(X \setminus f^{-1}(B)\) and \(f^{-1}(Y \setminus B)\text{?}\)
(b)
Let \(x \in f^{-1}(Y \setminus B)\text{.}\)
(i)
What does this tell us about \(f(x)\text{?}\)
(ii)
What can we conclude about the relationship between \(X \setminus f^{-1}(B)\) and \(f^{-1}(Y \setminus B)\text{?}\)
(c)
What is the relationship between \(X \setminus f^{-1}(B)\) and \(f^{-1}(Y \setminus B)\text{?}\)
Now we can consider the issue of continuity and closed sets.
Activity 10.5.
Let \(f\) be a function from a metric space \((X,d_X)\) to a metric space \((Y,d_Y)\text{.}\)
(a)
Assume that
\(f\) is continuous and that
\(C\) is a closed set in
\(Y\text{.}\) How does the result of
Activity 10.4 tell us that
\(f^{-1}(C)\) is closed in
\(X\text{?}\)
(b)
Now assume that
\(f^{-1}(C)\) is closed in
\(X\) whenever
\(C\) is closed in
\(Y\text{.}\) How does the result of
Activity 10.4 tell us that
\(f\) is a continuous function?
The result of
Activity 10.5 is summarized in the following theorem.
Theorem 10.5.
Let \(f\) be a function from a metric space \((X,d_X)\) to a metric space \((Y,d_Y)\text{.}\) Then \(f\) is continuous if and only if \(f^{-1}(C)\) is closed in \(X\) whenever \(C\) is a closed set in \(Y\text{.}\)
