Let \(p: E \to B\) be continuous and surjective. Suppose that \(U\) is an open set of \(B\) that is evenly covered by \(p\). Show that if \(U\) is connected, then the partition of \(p^{-1}(U)\) into slices is unique.
Solution
1. Slices over an evenly covered set:Proof:
1.1. By the definition of an evenly covered open set, there exists a partition of \(p^{-1}(U)\) into a collection of pairwise disjoint open sets \(\{V_\alpha\}_{\alpha \in A}\) in \(E\):\[p^{-1}(U) = \bigsqcup_{\alpha \in A} V_\alpha,\]such that for each \(\alpha \in A\), the restriction \(p|_{V_\alpha}: V_\alpha \to U\) is a homeomorphism.
1.2. The open sets \(V_\alpha\) are called the slices of \(p^{-1}(U)\).
2. Connectedness and clopen property of slices:Proof:
2.1. Because \(U\) is connected and \(p|_{V_\alpha}: V_\alpha \to U\) is a homeomorphism, the topological property of connectedness is preserved: each slice \(V_\alpha\) is a connected subspace of \(E\).
2.2. Since \(p^{-1}(U) \setminus V_\alpha = \bigcup_{\beta \neq \alpha} V_\beta\) is a union of open sets, it is open in \(p^{-1}(U)\).
2.3. Therefore each slice \(V_\alpha\) is both open and closed (clopen) in the subspace \(p^{-1}(U)\).
3. Characterization of slices as connected components:Proof:
3.1. Let \(C\) be any connected component of the subspace \(p^{-1}(U)\).
3.2. Pick a point \(x \in C\). Since the slices partition \(p^{-1}(U)\), there exists a unique \(\alpha \in A\) such that \(x \in V_\alpha\).
3.3. Since \(V_\alpha\) is clopen in \(p^{-1}(U)\), the intersection \(V_\alpha \cap C\) is clopen in the subspace \(C\).
3.4. Since \(x \in V_\alpha \cap C\), \(V_\alpha \cap C \neq \emptyset\). By connectedness of \(C\), the only non-empty clopen subset of \(C\) is \(C\) itself, so \(C \subseteq V_\alpha\).
3.5. Conversely, \(V_\alpha\) is a connected subset of \(p^{-1}(U)\) containing \(x\). Since \(C\) is the maximal connected subset containing \(x\), \(V_\alpha \subseteq C\).
3.6. Thus \(C = V_\alpha\).
4. Q.E.D.Proof:By step 3, the slices \(\{V_\alpha\}_{\alpha \in A}\) of \(p^{-1}(U)\) are precisely the connected components of the topological subspace \(p^{-1}(U)\). Because the partition of any topological space into its connected components is uniquely determined, the partition of \(p^{-1}(U)\) into slices is unique.