Section Compactness and Continuity
In our preview activity we learned about compactness — the analog of closed intervals from \(\R\) in topological spaces. Recall that a subset \(A\) of a topological space \(X\) is compact if every open cover of \(A\) has a finite sub-cover. As we will see, the definition of compactness is exactly what we need to ensure results of the type that continuous real-valued functions with domains in topological spaces attain maximum and minimum values on compact sets.
We might expect that compact sets have certain properties, but we must be careful which ones we assume.
Activity 17.2.
Let \(X = \{a,b,c,d\}\) and give \(X\) the topology \(\tau = \{\emptyset, \{a\}, \{b,c\}, \{a,b,c\}, X\}\text{.}\)
(a)
Explain why every finite subset of a topological space must be compact.
(b)
Find, if possible, a subset of \(X\) that is compact and open. If no such subset exists, explain why.
(c)
If \(A\) is a compact subset of \(X\text{,}\) must \(A\) be open? Explain.
(d)
Find, if possible, a subset of \(X\) that is compact and closed. If no such subset exists, explain why.
(e)
If \(A\) is a compact subset of \(X\text{,}\) must \(A\) be closed? Explain.
The message of
Activity 17.2 is that compactness by itself is not related to closed or open sets. We will see later, though, that in some reasonable circumstances, compact sets and closed sets are related. For the moment, we take a short detour and ask if compactness is a topological property.
Activity 17.3.
Let \((X, \tau_X)\) and \((Y, \tau_Y)\) be topological spaces, and let \(f: X \to Y\) be continuous. Assume that \(A\) is a compact subset of \(X\text{.}\) In this activity we want to determine if \(f(A)\) must be a compact subset of \(Y\text{.}\)
(a)
What do we need to show to prove that \(f(A)\) is a compact subset of \(Y\text{?}\) Where do we start?
(b)
If we have an open cover of \(f(A)\) in \(Y\text{,}\) how can we find an open cover \(\{U_{\alpha}\}\) for \(A\text{?}\) Be sure to verify that what you claim is actually an open cover of \(A\text{.}\)
(c)
What do we know about any open cover of \(A\text{?}\)
(d)
Complete the proof of the following theorem.
Theorem 17.4.
Let \((X, \tau_X)\) and \((Y, \tau_Y)\) be topological spaces, and let \(f: X \to Y\) be continuous. If \(A\) is a compact subset of \(X\text{,}\) then \(f(A)\) is a compact subset of \(Y\text{.}\)
A consequence of
Activity 17.3 is that compactness is a topological property.
Corollary 17.5.
Let \((X, \tau_X)\) and \((Y, \tau_Y)\) be homeomorphic topological spaces. Then a subset \(A\) of \(X\) is compact if and only if \(f(A)\) is compact in \(Y\text{.}\)
