Let \(X\) be a space; let \(\mathcal{A}\) be an open covering of \(X\). Under what conditions does there exist an open covering \(\mathcal{B}\) of \(X\) refining \(\mathcal{A}\) such that for each pair \(B, B'\) of elements of \(\mathcal{B}\) that have nonempty intersection, the union \(B \cup B'\) lies in an element of \(\mathcal{A}\)?
Show that such a covering \(\mathcal{B}\) exists if \(X\) is metrizable. [Hint: Choose \(\epsilon(x)\) so that \(B(x, 3\epsilon(x))\) lies in an element of \(\mathcal{A}\). Let \(\mathcal{B}\) consist of the open sets \(B(x, \epsilon(x))\).]
Show that such a covering exists if \(X\) is compact Hausdorff. [Hint: Let \(A_1, \ldots, A_n\) be a finite subcollection of \(\mathcal{A}\) that covers \(X\). Choose an open covering \(C_1, \ldots, C_n\) of \(X\) such that \(\overline{C}_i \subset A_i\) for each \(i\). For each nonempty subset \(J\) of \(\ts{1, \ldots, n}\), consider the set
\[ B_J = \bigcap_{j \in J} A_j - \bigcup_{j \notin J} \overline{C}_j.] \]
Solution
Goal: Prove the existence of a star-fine open refinement \(\mathcal{B}\) of an open covering \(\mathcal{A}\) (satisfying \(B \cap B' \neq \varnothing \implies B \cup B' \subseteq A\) for some \(A \in \mathcal{A}\)) when \(X\) is metrizable or compact Hausdorff.
1. Part (a): Case where \(X\) is metrizable.Proof:
1.1. Let \(d\) be a compatible metric on \(X\).
1.2. For each \(x \in X\), choose \(A_x \in \mathcal{A}\) containing \(x\). Since \(A_x\) is open, there is \(r_x > 0\) such that the metric ball \(B(x, r_x) \subseteq A_x\).
1.3. Set \(\varepsilon(x) = r_x / 3 > 0\), so \(B(x, 3\varepsilon(x)) \subseteq A_x \in \mathcal{A}\).
1.4. Define \(\mathcal{B} = \{B(x, \varepsilon(x)) \mid x \in X\}\). Since \(x \in B(x, \varepsilon(x))\), \(\mathcal{B}\) is an open covering of \(X\).
1.5. Suppose \(B_1 = B(x_1, \varepsilon(x_1))\) and \(B_2 = B(x_2, \varepsilon(x_2))\) have non-empty intersection, with \(z \in B_1 \cap B_2\).
1.6. Without loss of generality, assume \(\varepsilon(x_1) \ge \varepsilon(x_2)\).
1.7. For any \(y \in B_2\), \(d(y, x_1) \le d(y, x_2) + d(x_2, z) + d(z, x_1) < \varepsilon(x_2) + \varepsilon(x_2) + \varepsilon(x_1) \le 3\varepsilon(x_1)\).
1.8. For any \(y \in B_1\), \(d(y, x_1) < \varepsilon(x_1) < 3\varepsilon(x_1)\).
1.9. Thus \(B_1 \cup B_2 \subseteq B(x_1, 3\varepsilon(x_1)) \subseteq A_{x_1} \in \mathcal{A}\).
2. Part (b): Case where \(X\) is compact Hausdorff.Proof:
2.1. By compactness of \(X\), extract a finite subcover \(\{A_1, \dots, A_n\} \subseteq \mathcal{A}\).
2.2. Because compact Hausdorff spaces are normal, the Shrinking Lemma gives an open cover \(\{C_1, \dots, C_n\}\) of \(X\) with \(\overline{C}_i \subseteq A_i\) for each \(i = 1, \dots, n\).
2.3. For each non-empty subset \(J \subseteq \{1, \dots, n\}\), define:\[B_J = \left( \bigcap_{j \in J} A_j \right) \setminus \left( \bigcup_{k \notin J} \overline{C}_k \right) = \left( \bigcap_{j \in J} A_j \right) \cap \left( \bigcap_{k \notin J} (X \setminus \overline{C}_k) \right).\]
2.4. Each \(B_J\) is open, being a finite intersection of open sets.
2.5. \(\mathcal{B} = \{B_J\}\) covers \(X\): For any \(x \in X\), let \(J(x) = \{i \mid x \in \overline{C}_i\}\). Since \(\{C_i\}\) covers \(X\), \(J(x) \neq \varnothing\). By definition, \(x \in \overline{C}_j \subseteq A_j\) for \(j \in J(x)\) and \(x \notin \overline{C}_k\) for \(k \notin J(x)\), so \(x \in B_{J(x)}\).
2.6. Star refinement property: Suppose \(B_J \cap B_K \neq \varnothing\), and let \(x \in B_J \cap B_K\).
2.7. Since \(\{C_1, \dots, C_n\}\) covers \(X\), \(x \in C_{j_0} \subseteq \overline{C}_{j_0}\) for some \(j_0\).
2.8. Because \(x \in B_J\), \(x \notin \overline{C}_k\) for all \(k \notin J\), which forces \(j_0 \in J\).
2.9. Symmetrically, \(x \in B_K\) forces \(j_0 \in K\).
2.10. Thus \(j_0 \in J \cap K\).
2.11. By definition of \(B_J\) and \(B_K\), \(B_J \subseteq A_{j_0}\) and \(B_K \subseteq A_{j_0}\), so:\[B_J \cup B_K \subseteq A_{j_0} \in \mathcal{A}.\]
3. Conclusion:Such an open star-refinement \(\mathcal{B}\) exists whenever \(X\) is metrizable or compact Hausdorff (and more generally, whenever \(X\) is paracompact). Q.E.D.