Show that there are only countably many homotopy types of finite CW complexes.
Solution
Hatcher’s Theorem 2C.5 says that every finite CW complex is homotopy equivalent to a finite simplicial complex. It therefore suffices to count finite simplicial complexes up to isomorphism.
1. For each positive integer \(N\), there are only finitely many simplicial complexes with vertex set exactly\[ \{1,\ldots,N\}. \]
Such a simplicial complex is a collection of nonempty subsets of \(\{1,\ldots,N\}\) closed under passage to nonempty subsets. There are only finitely many subsets of the finite power set, so only finitely many possible complexes.
2. Hence there are only countably many isomorphism classes of finite simplicial complexes.
1. Take the union over \(N\in\mathbb N\).
3. Consequently there are only countably many homotopy types of finite CW complexes.
2. Passing from isomorphism classes to homotopy types can only identify elements, not create new ones.
Thus \[ \boxed{\text{the set of homotopy types of finite CW complexes is countable}.} \]