I'm reading a paper from Y. Ünlü called Lattices of compactifications of Tychonoff spaces. I've bumped into some definitions that I've never seen; while I find most of them understandable, there's one that I can't break down.
Definition 1. A decomposition $\alpha$ of a space $X$ is a partition of $X$ into compact sets.
So far, this one is easy to understand.
Definition 2. A decomposition $\alpha$ of a set $X$ is called upper semicontinuous if for $K \in \alpha$, $K \subset V$ and $V$ is open in $X$, then there is an open set $W$ in $X$ such that $K \subset W \subset V$ and $W = \bigcup \{L \in \alpha\ \textrm{s.t.}\ L \cap W \neq \emptyset\}$. A set $W$ satisfying this condition is called $\alpha$-saturated.
It's here where I come into trouble. In $W = \bigcup \{L \in \alpha\ \textrm{s.t.}\ W \cap L \neq \emptyset\}$ how can we express $W$ in terms of itself, if it hasn't already been defined?
Could you please clarify this definition (maybe it's a typo and the author wanted to write $V$ instead of $W$, or maybe I'm not good enough to get it) and give me some equivalent statement?
A better formulation of the notion of upper semicontinuous decompositions is to demand
See this survey for more explanation.
Then if $\alpha$ satisfies this and we have (as in your situation) $K \in \alpha, K \subseteq V$, we take $W$ as promised, which is open (and clearly contains $\alpha$) and if $L \in \alpha$ intersects $W$, then $L \subseteq W$ (because we have a partition: some $x \in L$ is contained in an $L'$ that sits inside $W$ (by definition of $W$) but $L=L'$ as intersecting means equality in a partition.), and $W$ is saturated.
And no, the definition as stated is just the definition of being saturated not the definition of $W$ itself. A set $S$ (in general) is saturated for a partition $\alpha$, iff $$\forall K \in \alpha: (K \cap S \neq \emptyset) \to K \subseteq S$$
(so if you have a point of a member you have the whole member; " if you eat them a bit, you eat them completely", to put it weirdly.)