Prove the following.
Theorem. Let \(X\) be a space that is path connected, locally path connected, and semilocally simply connected. If \(X\) is compact Hausdorff, then \(\pi_1(X, x_0)\) is finitely generated, and hence countable.
Proof. Repeat the proof outlined in E-QVJ8J, choosing \(\mathcal{B}\) to be finite. One has the equation
\[ [f] = [g_1] * \dots * [g_n], \]
as before. Choose, for each \(x \in X\), a path \(\beta_x\) from \(x_0\) to \(x\); let \(\beta_{x_0}\) be the constant path. If \(g = g(B, B')\), define
\[ L(g) = \beta_x * (g * \overline{\beta}_y), \]
where \(x = p(B)\) and \(y = p(B')\). Show that
\[ [f] = [L(g_1)] * \dots * [L(g_n)]. \]
Solution
Goal: Prove that if \(X\) is a compact, path-connected, locally path-connected, and semilocally simply connected space, then its fundamental group \(\pi_1(X, x_0)\) is finitely generated (and countable).
1. Finite cover by canonical open sets:Proof:
1.1. By local path-connectedness and semilocal simple-connectedness, each \(x \in X\) has an open path-connected neighborhood \(U_x\) such that the inclusion homomorphism \(\pi_1(U_x, x) \to \pi_1(X, x)\) is trivial.
1.2. The collection \(\{U_x\}_{x \in X}\) forms an open cover of \(X\).
1.3. Because \(X\) is compact, there exists a finite subcover \(\mathcal{B} = \{B_1, \dots, B_m\}\).
1.4. For each \(B_i \in \mathcal{B}\), choose a designated basepoint \(p(B_i) \in B_i\), and fix a path \(\beta_{p(B_i)}\) in \(X\) from \(x_0\) to \(p(B_i)\) (with \(\beta_{x_0}\) the constant path if \(p(B_i) = x_0\)).
2. Finite generating family of loops:Proof:
2.1. For any pair \((B, B') \in \mathcal{B} \times \mathcal{B}\) with \(B \cap B' \neq \varnothing\), choose a point \(q \in B \cap B'\).
2.2. Choose a path \(\alpha\) in \(B\) from \(p(B)\) to \(q\), and a path \(\alpha'\) in \(B'\) from \(p(B')\) to \(q\).
2.3. Define \(g(B, B') = \alpha * \overline{\alpha'}\), which is a path in \(B \cup B'\) from \(p(B)\) to \(p(B')\).
2.4. Define the based loop:\[L(g(B, B')) = \beta_{p(B)} * g(B, B') * \overline{\beta}_{p(B')}.\]
2.5. Because \(\mathcal{B}\) is a finite set of \(m\) open sets, the set of non-empty pairs is finite (at most \(m^2\)).
2.6. Hence \(\mathcal{S} = \{ [L(g(B, B'))] \mid B, B' \in \mathcal{B}, B \cap B' \neq \varnothing \}\) is a finite subset of \(\pi_1(X, x_0)\).
3. Decomposition and factorization of an arbitrary loop \(f\):Proof:
3.1. Let \(f: [0, 1] \to X\) be an arbitrary continuous loop based at \(x_0\).
3.2. By the Lebesgue Covering Lemma, subdivide \([0, 1]\) into \(0 = t_0 < t_1 < \dots < t_n = 1\) such that \(f([t_{k-1}, t_k]) \subseteq B_{i_k}\) for some \(B_{i_k} \in \mathcal{B}\).
3.3. For each \(k\), \(f(t_k) \in B_{i_k} \cap B_{i_{k+1}}\), so let \(g_k = g(B_{i_k}, B_{i_{k+1}})\).
3.4. Because the inclusion \(\pi_1(B_i) \to \pi_1(X)\) is trivial, paths in \(B_i\) with identical endpoints are path-homotopic in \(X\), giving the homotopy \([f] = [g_1 * g_2 * \dots * g_n]\).
3.5. Inserting inverse pairs \(\overline{\beta}_{p(B_{i_k})} * \beta_{p(B_{i_k})} \simeq c_{p(B_{i_k})}\) between successive segments:\[[f] = [(\beta_{x_0} * g_1 * \overline{\beta}_{p(B_{i_2})}) * (\beta_{p(B_{i_2})} * g_2 * \overline{\beta}_{p(B_{i_3})}) * \dots * (\beta_{p(B_{i_n})} * g_n * \overline{\beta}_{x_0})]\]\[= [L(g_1)] * [L(g_2)] * \dots * [L(g_n)].\]
4. Conclusion:Every \([f] \in \pi_1(X, x_0)\) is a product of elements from the finite set \(\mathcal{S}\).Thus \(\pi_1(X, x_0)\) is finitely generated, and hence countable. Q.E.D.