# Star refinement and paracompactness

If $$\mathcal{U}$$ is a cover of a space $$X$$ and $$S\subseteq X$$, define the star of $$S$$ with respect to $$\mathcal{U}$$ as $$\text{st}(S,\mathcal{U})=\bigcup\{U\in\mathcal{U}\mid S\cap U\neq\emptyset\}$$.

1. According to wikipedia, a Hausdorff space is paracompact iff it is fully normal; it is also defined there that a space is fully normal if every open cover has an open star refinement. In the same page it says a cover $$\mathcal{U}$$ is a star refinement of $$\mathcal{V}$$ if for any $$x\in X$$, $$\text{st}(\{x\},\mathcal{U})\subseteq V$$ for some $$V\in\mathcal{V}$$. However the wikipage for star refinement says it should be for any $$U\in\mathcal{U}$$, $$\text{st}(U,\mathcal{U})\subseteq V$$ for some $$V\in\mathcal{V}$$, which is a priori stronger. Do these give the same definition? If not which one is the correct definition of fully normal spaces?

2. Consider the following property: any open cover $$\mathcal{V}$$ has an open refinement $$\mathcal{U}$$ such that any $$U\in\mathcal{U}$$ intersects only finitely many other members of $$\mathcal{U}$$. This property easily implies paracompactness. Is the converse true (maybe under mild assumption)? If not does this property has a name?

By fact 1 in this note it follows that the notion of every cover having a star-refinement (with the $$U$$ as first component) is equivalent to having a barycentric open refinement for each open cover (with $$\{x\}$$ as first component).
A cover such that every element intersects only finitely many other elements is calls star-finite and the property that every open cover has a star-finite open refinement is called strongly paracompact and indeed implies paracompact (but not always conversely). Asking for a point-finite refinement instead (every point is in finitely many members) is called weakly paracompact (or metacompact in some texts) and is strictly weaker than paracompactness. E.g. A metric space is always paracompact but only strongly paracompact when we can embed it into $$[0,1]^\omega \times D^\omega$$ for some discrete space $$D$$ (Morita's theorem).