MathOverflow

I'm struggling to understand an argument in the book "Weil Conjectures, Perverse Sheaves, and $l$-adic Fourier Transform" by Kiehl and Weissauer.
In Theorem I.6.1, they prove (in specific cases) that the $i$-th cohomology of a pure sheaf of weight $w$ has weight $w+i$. I'm confused by their argument on the second half of page $47$ where they treat the special case of a geometrically constant sheaf...

Let $G=(V,E)$ be a finite graph together with a fixed total ordering of its vertices. Consider the following greedy procedure.
Let $R_0=V$. At step $k$, choose the first vertex $r_k \in R_k$ in the prescribed order. Form the block:
$$B_k = N[r_k] \cap R_k$$...

I have a question about a definition in the immersion theorem of Lees and Lashof: in Lashof's paper "The immersion approach to triangulation and smoothing" he proves that the differential map from the space of immersions of an open manifold to the space...

Recall that a morphism $f\colon X\to Y$ of locales is said to be open when its inverse image part $f^*\colon \mathcal{O}(Y) \to \mathcal{O}(X)$ (where $\mathcal{O}(X)$ denotes the frame of opens which defines the locale $X$) has a left adjoint $f_!\colon...