Every open cover of a compact Hausdorff space has a finite open star-refinement

Let $\mathcal{V}$ be the family of all open sets $V$ such that $\bar{V}\subseteq U$ for some $U\in \mathcal{U}$. This family covers $X$. Indeed, for $x\in U\in \mathcal{U}$, normality separates the closed sets $\{x\}$ and $X\setminus U$ by disjoint open sets; the open set containing $x$ has closure contained in $U$. This definition uses no choices indexed by $X$.

We record a finite shrinking construction. Given a finite open cover $A_0,\dots ,A_{m-1}$, recursively put $F_i=X\setminus ({\bigcup }_{j<i}{B}_j\cup {\bigcup }_{j>i}{A}_j)\mathrm{.}$ The earlier covering clauses imply $F_i\subseteq A_i$. Normality separates $F_i$ from $X\setminus A_i$, giving an open $B_i$ with $F_i\subseteq B_i\subseteq \overline{B_i}\subseteq A_i$. At the last stage the $B_i$ cover $X$. Thus every finite open cover has an open shrinking whose closures remain in the original members.

Compactness gives a finite subcover $V_0,\dots ,V_{n-1}$ of $\mathcal{V}$. Finite choice supplies $U_i\in \mathcal{U}$ with $\overline{V_i}\subseteq U_i$ for each $i<n$.

From a finite cover $A_i$ and an open shrinking $B_i$ as in step 1.2, form, for each nonempty $S\subseteq \{0,\dots ,m-1\}$, $W_S=\left({\bigcap }_{i\in S}{A}_i\right)\setminus \left({\bigcup }_{j\mathrm{\notin }S}\overline{B_j}\right),$ discarding empty members. These finitely many sets are open and cover $X$: at $x$, take $S=\{i:x\in A_i\}$. Moreover, choose $k$ with $x\in B_k$. Every $W_S$ containing $x$ has $k\in S$, so $W_S\subseteq A_k$. Hence the point-star $\mathrm{St}(x,\mathcal{W})$ lies in $A_k$. Call this a barycentric refinement of $(A_i)$.

Apply step 2.2 to the finite cover $(U_i)$ and its shrinking $(V_i)$ from step 2.1, obtaining a finite open barycentric refinement $\mathcal{W}$ of $\mathcal{U}$. Apply steps 1.2 and 2.2 again to $\mathcal{W}$, obtaining a finite open barycentric refinement $\mathcal{Z}$ of $\mathcal{W}$.

The cover $\mathcal{Z}$ star-refines $\mathcal{U}$. Fix $Z_0\in \mathcal{Z}$ and $x\in Z_0$. Barycentricity of $\mathcal{W}$ gives $U\in \mathcal{U}$ with $\mathrm{St}(x,\mathcal{W})\subseteq U$. If $Z\in \mathcal{Z}$ meets $Z_0$ at $y$, barycentricity of $\mathcal{Z}$ gives $W_y\in \mathcal{W}$ containing $\mathrm{St}(y,\mathcal{Z})$. Both $Z_0$ and $Z$ lie in $W_y$, and $x\in W_y$, so $Z\subseteq W_y\subseteq \mathrm{St}(x,\mathcal{W})\subseteq U$. Thus $\mathrm{St}(Z_0,\mathcal{Z})\subseteq U$.

The finite open cover $\mathcal{Z}$ is therefore a star-refinement of the original cover.