A Complete Spread I am reading R. Fox's "Covering Spaces with Singularities", which deals with a careful definition of branched covering spaces. I am having trouble understanding the exact definition and importance of the "completion" of a "spread", which is something like a pre-branched covering. It's a purely topological thing - here are the definitions.
A map $g:Y\to Z$ is a spread if the (connected) components of the inverse images of the open sets of $Z$ form a basis for $Y$ ($Y$ is $T_1$, $Z$ is locally connected $T_1$).
A spread is complete if for every $z\in Z$ the following is true: If for every neighborhood $W$ of $z$ there is a selected component $V$ of $g^{-1}(W)$ such that $V_1\subset V_2$ whenever $W_1\subset W_2$, then $\cap_W V$ is non-vacuous (and is therefore a point).
The real problem is understanding what "$\cup_W V$" means. Is this the intersection of all $V_i$ that satisfy $V_i\subset V_{i+1}$ when $W_i\subset W_{i+1}$? If so, what is the point of the definition, just that one can take open sets to be as small as we want? In this case, I suppose a non-complete spread is one, say, that is missing a point $p\in V_i$ so that $\cap_W V = \emptyset$?
Or, is "$\cap_W V$" over the entirety of $W$, all components of $V$? This seems to make less sense because the $V$ are supposed to be connected components so naturally $\cap _W V = 0$.
Can someone clarify this? It just seems like maybe this is indirectly saying something I am not understanding.
 A: I have found some more details in a recent paper by Montesinos (2005), Branched Coverings After Fox. Essentially I was reasonably correct, but here are some details given in that paper.
For a spread $g:Y\to Z$ and a set of open neighborhoods $\mathcal{E}(z)$ of $z\in Z$, there is a directed set $(\mathcal{E}(z),\geq)$ defined by $W_2 \geq W_1$ if $W_2 \subset W_1$ for $W_1,W_2\in \mathcal{E}(z)$. Define $Y_W$ as the space obtained by identifying each connected component of $g^{-1}(W)$ to a single point. The limit obtained in this manner from the directed set, $Y_z$, are threads over $z$.
For each thread $y_z=\{y_z(W)\}$ over $z$ (which is a union of a single connected component of $g^{-1}(W)$), the intersection
$$\bigcap_{W\in\mathcal{E}(z)}y_z(W)$$
is a single point, at most. This comes from the points in the fiber having the discrete topology and Hausdorff.
If every thread $y_z=\{y_z(W)\}$ has a single point in its intersection, it is complete.
So essentially, a complete spread is a refined notion of an unbranched covering space. Each point in the base has separable components in the cover. Fox will go on to show that every spread is completable, showing that every branched covering is a manifold.
